4.6: Gradient, Divergence, Curl, and Laplacian
In this final section we will establish some relationships between the gradient, divergence and curl, and we will also introduce a new quantity called the Laplacian . We will then show how to write these quantities in cylindrical and spherical coordinates.
Gradient
For a real-valued function \(f (x, y, z)\) on \(\mathbb{R}^ 3\), the gradient \(∇f (x, y, z)\) is a vector-valued function on \(\mathbb{R}^ 3\), that is, its value at a point \((x, y, z)\) is the vector
\[\nonumber ∇f (x, y, z) = \left ( \dfrac{∂f}{ ∂x} , \dfrac{∂f}{ ∂y} , \dfrac{∂f}{ ∂z} \right ) = \dfrac{∂f}{ ∂x}\textbf{i} + \dfrac{∂f}{ ∂y}\textbf{j} + \dfrac{∂f}{ ∂z}\textbf{k} \]
in \(\mathbb{R}^ 3\), where each of the partial derivatives is evaluated at the point \((x, y, z)\). So in this way, you can think of the symbol \(∇\) as being “applied” to a real-valued function \(f\) to produce a vector \(∇f\).
It turns out that the divergence and curl can also be expressed in terms of the symbol \(∇\). This is done by thinking of \(∇\) as a vector in \(\mathbb{R}^ 3\), namely
\[∇ = \dfrac{∂}{ ∂x}\textbf{i} + \dfrac{∂}{ ∂y}\textbf{j} + \dfrac{∂}{ ∂z}\textbf{k} .\label{Eq4.51}\]
Here, the symbols \(\dfrac{∂}{ ∂x} ,\, \dfrac{∂}{∂y} \text{ and }\dfrac{∂}{ ∂z}\) are to be thought of as “partial derivative operators” that will get “applied” to a real-valued function, say \(f (x, y, z)\), to produce the partial derivatives \(\dfrac{∂f}{ ∂x} ,\, \dfrac{ ∂f}{ ∂y}\text{ and }\dfrac{∂f}{ ∂z}\). For instance, \(\dfrac{∂}{ ∂x}\) “applied” to \(f (x, y, z) \text{ produces }\dfrac{∂f}{ ∂x}\).
Is \(∇\) really a vector? Strictly speaking, no, since \(\dfrac{∂}{ ∂x} ,\, \dfrac{∂}{ ∂y} \text{ and }\dfrac{∂}{ ∂z}\) are not actual numbers. But it helps to think of \(∇\) as a vector, especially with the divergence and curl, as we will soon see. The process of “applying” \(\dfrac{∂}{ ∂x} ,\, \dfrac{∂}{ ∂y} ,\, \dfrac{∂}{ ∂z}\) to a real-valued function \(f (x, y, z)\) is normally thought of as multiplying the quantities:
\[\nonumber \left ( \dfrac{∂}{ ∂x} \right ) (f ) = \dfrac{∂f}{ ∂x},\, \left ( \dfrac{∂}{ ∂y} \right ) (f ) = \dfrac{∂f}{ ∂y} ,\, \left ( \dfrac{∂}{ ∂z} \right ) (f ) = \dfrac{∂f}{ ∂z}\]
For this reason, \(∇\) is often referred to as the “del operator”, since it “operates” on functions.
Divergence
For example, it is often convenient to write the divergence div f as \(∇ \cdot \textbf{f}\), since for a vector field \(\textbf{f}(x, y, z) = f_1(x, y, z)\textbf{i}+ f_2(x, y, z)\textbf{j}+ f_3(x, y, z)\textbf{k}\), the dot product of f with \(∇\) (thought of as a vector) makes sense:
\[\nonumber \begin{align} ∇·\textbf{f} &= \left ( \dfrac{∂}{ ∂x}\textbf{i} + \dfrac{∂}{ ∂y}\textbf{j} + \dfrac{∂}{ ∂z}\textbf{k} \right ) · (f_1(x, y, z)\textbf{i} + f_2(x, y, z)\textbf{j} + f_3(x, y, z)\textbf{k}) \\[4pt] \nonumber &= \left ( \dfrac{∂}{ ∂x} \right ) (f_1) + \left ( \dfrac{∂}{ ∂y} \right ) (f_2) + \left ( \dfrac{∂}{ ∂z} \right ) (f_3) \\[4pt] \nonumber &= \dfrac{∂f_1}{ ∂x} + \dfrac{∂f_2}{ ∂y} + \dfrac{∂f_3}{ ∂z} \\[4pt] \nonumber &= \text{div }\textbf{f} \\[4pt] \end{align}\]
We can also write curl f in terms of \(∇\), namely as \(∇ × \textbf{f}\), since for a vector field \(\textbf{f}(x, y, z) = P(x, y, z)\textbf{i}+Q(x, y, z)\textbf{j}+ R(x, y, z)\textbf{k}\), we have:
\[\nonumber \begin{align} ∇ × \textbf{f} &= \begin{vmatrix} \textbf{i} & \textbf{j} & \textbf{k} \\[4pt] \dfrac{∂}{ ∂x} & \dfrac{∂}{ ∂y} & \dfrac{∂}{ ∂z} \\[4pt] \nonumber P(x, y, z) & Q(x, y, z) & R(x, y, z) \\[4pt] \end{vmatrix} \\[4pt] \nonumber &=\left ( \dfrac{∂R}{ ∂y} − \dfrac{∂Q}{ ∂z}\right )\textbf{i} − \left ( \dfrac{∂R}{ ∂x} − \dfrac{∂P}{ ∂z} \right )\textbf{j} + \left ( \dfrac{∂Q}{ ∂x} − \dfrac{∂P}{ ∂y} \right )\textbf{k} \\[4pt] \nonumber &= \left ( \dfrac{∂R}{ ∂y} − \dfrac{∂Q}{ ∂z}\right )\textbf{i} + \left (\dfrac{∂P}{ ∂z}-\dfrac{∂R}{ ∂x} \right )\textbf{j} + \left ( \dfrac{∂Q}{ ∂x} − \dfrac{∂P}{ ∂y} \right )\textbf{k} \\[4pt] \nonumber &=\text{curl }\textbf{f} \\[4pt] \end{align}\]
For a real-valued function \(f (x, y, z)\), the gradient \(∇f (x, y, z) = \dfrac{∂f}{ ∂x}\textbf{i} + \dfrac{∂f}{ ∂y}\textbf{j} + \dfrac{∂f}{ ∂z}\textbf{k}\) is a vector field, so we can take its divergence:
\[\nonumber \begin{align} \text{div }∇f &= ∇· ∇f \\[4pt] \nonumber &=\left ( \dfrac{∂}{ ∂x}\textbf{i} + \dfrac{∂}{ ∂y}\textbf{j} + \dfrac{∂}{ ∂z}\textbf{k} \right ) · \left ( \dfrac{∂f}{ ∂x}\textbf{i} + \dfrac{∂f}{ ∂y}\textbf{j} + \dfrac{∂f}{ ∂z}\textbf{k} \right ) \\[4pt] \nonumber &=\dfrac{∂}{ ∂x} \left ( \dfrac{∂f}{ ∂x} \right ) + \dfrac{∂}{ ∂y} \left ( \dfrac{∂f}{ ∂y} \right ) + \dfrac{∂}{ ∂z} \left ( \dfrac{∂f}{ ∂z} \right ) \\[4pt] \nonumber &= \dfrac{∂^2 f}{ ∂x^ 2} + \dfrac{∂^ 2 f}{ ∂y^ 2} + \dfrac{∂^ 2 f}{ ∂z^ 2} \\[4pt] \end{align}\]
Note that this is a real-valued function, to which we will give a special name:
Definition 4.7: Laplacian
For a real-valued function \(f (x, y, z)\), the Laplacian of \(f\), denoted by \(∆f\), is given by
\[∆f (x, y, z) = ∇· ∇f = \dfrac{∂^ 2 f}{ ∂x^ 2} + \dfrac{∂^ 2 f}{ ∂y^ 2} + \dfrac{∂^ 2 f}{ ∂z^ 2} .\label{Eq4.52}\]
Often the notation \(∇^2 f\) is used for the Laplacian instead of \(∆f\), using the convention \(∇^ 2 = ∇· ∇\).
Example 4.17
Let \(\textbf{r}(x, y, z) = x\textbf{i}+y\textbf{j}+z\textbf{k}\) be the position vector field on \(\mathbb{R}^ 3\). Then \(\lVert \textbf{r}(x, y, z)\rVert ^2 = \textbf{r} · \textbf{r} = x^ 2 + y^ 2 + z^ 2\) is a real-valued function. Find
- the gradient of \(\lVert \textbf{r} \rVert ^2\)
- the divergence of \(\textbf{r}\)
- the curl of \(\textbf{r}\)
- the Laplacian of \(\lVert \textbf{r} \rVert ^2\)
Solution:
(a) \(∇ \rVert \textbf{r}\rVert ^2 = 2x\textbf{i}+2y\textbf{j}+2z\textbf{k} = 2\textbf{r}\)
(b) \(∇· \textbf{r} = \dfrac{∂}{ ∂x} (x)+ \dfrac{∂}{ ∂y} (y)+ \dfrac{∂}{ ∂z} (z) = 1+1+1 = 3\)
(c)
\[\nonumber ∇ × \textbf{r} = \begin{vmatrix} \textbf{i} & \textbf{j} & \textbf{k} \\[4pt] \dfrac{∂}{ ∂x} & \dfrac{∂}{ ∂y} & \dfrac{∂}{ ∂z} \\[4pt] x & y & z \\[4pt] \end{vmatrix} = (0−0)\textbf{i} − (0−0)\textbf{j} + (0−0)\textbf{k} = \textbf{0} \]
(d) \(∆ \lVert \textbf{r} \rVert ^2 = \dfrac{∂^ 2}{ ∂x^ 2} (x^ 2 + y^ 2 + z^ 2 )+ \dfrac{∂^ 2}{ ∂y^ 2} (x^ 2 + y^ 2 + z^ 2 )+ \dfrac{∂^ 2}{ ∂z^ 2} (x^ 2 + y^ 2 + z^ 2 ) = 2+2+2 = 6\)
Note that we could have calculated \(∆\lVert \textbf{r} \rVert ^2\) another way, using the \(∇\) notation along with parts (a) and (b):
\[\nonumber ∆\lVert \textbf{r} \rVert ^2 = ∇· ∇ \lVert \textbf{r} \rVert ^2 = ∇· 2\textbf{r} = 2∇· \textbf{r} = 2(3) = 6\]
Notice that in Example 4.17 if we take the curl of the gradient of \(\lVert \textbf{r} \rVert ^2\) we get
\[\nonumber ∇ × (∇ \rVert \textbf{r}\rVert ^2 ) = ∇ × 2\textbf{r} = 2∇ × \textbf{r} = 2\textbf{0} = \textbf{0} .\]
The following theorem shows that this will be the case in general:
Theorem 4.15.
For any smooth real-valued function \(f (x, y, z), ∇ × (∇f ) = \textbf{0}\).
Proof
We see by the smoothness of f that
\[\begin{align} ∇ × (∇f ) &= \begin{vmatrix} \textbf{i} & \textbf{j} & \textbf{k} \\[4pt] \dfrac{∂}{ ∂x} & \dfrac{∂}{ ∂y} & \dfrac{∂}{ ∂z} \\[4pt] \dfrac{∂f}{ ∂x} & \dfrac{∂f}{ ∂y} & \dfrac{∂f}{ ∂z} \\[4pt] \end{vmatrix} \\[4pt] &= \left ( \dfrac{∂^ 2 f}{ ∂y∂z} − \dfrac{∂^ 2 f}{ ∂z∂y} \right )\textbf{i} − \left ( \dfrac{∂^ 2 f}{ ∂x∂z} − \dfrac{∂^ 2 f}{ ∂z∂x} \right )\textbf{j} + \left ( \dfrac{∂^ 2 f}{ ∂x∂y} − \dfrac{∂^ 2 f}{ ∂y∂x} \right )\textbf{k} = \textbf{0} ,\\[4pt] \end{align}\]
since the mixed partial derivatives in each component are equal.
\(\square\)
Corollary 4.16
If a vector field \(f(x, y, z)\) has a potential, then curl \(\textbf{f} = \textbf{0}\).
Another way of stating Theorem 4.15 is that gradients are irrotational. Also, notice that in Example 4.17 if we take the divergence of the curl of r we trivially get
\[∇· (∇ × \textbf{r}) = ∇· \textbf{0} = 0 .\]
The following theorem shows that this will be the case in general:
Theorem 4.17.
For any smooth vector field \(\textbf{f}(x, y, z), ∇· (∇ × \textbf{f}) = 0.\)
The proof is straightforward and left as an exercise for the reader.
Corollary 4.18
The flux of the curl of a smooth vector field \(f(x, y, z)\) through any closed surface is zero.
Proof : Let \(Σ\) be a closed surface which bounds a solid \(S\). The flux of \(∇ × \textbf{f}\) through \(Σ\) is
\[\begin{align} \iint\limits_Σ (∇ × \textbf{f})· dσ &= \iiint\limits_S ∇· (∇ × \textbf{f})\, dV \text{ (by the Divergence Theorem)} \\[4pt] &= \iiint\limits_S 0\, dV \text{ (by Theorem 4.17)} \\[4pt] &= 0 \\[4pt] \end{align}\]
\(\tag{\(\textbf{QED}\)}\)
There is another method for proving Theorem 4.15 which can be useful, and is often used in physics. Namely, if the surface integral \(\iint\limits_Σ f (x, y, z)\,dσ = 0\) for all surfaces \(Σ\) in some solid region (usually all of \(\mathbb{R}^ 3\) ), then we must have \(f (x, y, z) = 0\) throughout that region. The proof is not trivial, and physicists do not usually bother to prove it. But the result is true, and can also be applied to double and triple integrals.
For instance, to prove Theorem 4.15, assume that \(f (x, y, z)\) is a smooth real-valued function on \(\mathbb{R}^ 3\). Let \(C\) be a simple closed curve in \(\mathbb{R}^ 3\) and let \(Σ\) be any capping surface for \(C\) (i.e. \(Σ\) is orientable and its boundary is \(C\)). Since \(∇f\) is a vector field, then
\[\nonumber \begin{align} \iint\limits_Σ (∇ × (∇f ))· \textbf{n}\,dσ &= \oint_C ∇f · d\textbf{r} \text{ by Stokes’ Theorem, so} \\[4pt] \nonumber &= 0 \text{ by Corollary 4.13.} \\[4pt] \end{align}\]
Since the choice of \(Σ\) was arbitrary, then we must have \((∇×(∇f ))·\textbf{n} = 0\) throughout \(\mathbb{R}^ 3\), where n is any unit vector. Using i , j and k in place of n , we see that we must have \(∇ × (∇f ) = \textbf{0}\) in \(\mathbb{R}^ 3\), which completes the proof.
Example 4.18
A system of electric charges has a charge density \(ρ(x, y, z)\) and produces an electrostatic field \(\textbf{E}(x, y, z)\) at points \((x, y, z)\) in space. Gauss’ Law states that
\[\nonumber \iint\limits_Σ \textbf{E}· dσ = 4π \iiint\limits_S ρ \,dV\]
for any closed surface \(Σ\) which encloses the charges, with \(S\) being the solid region enclosed by \(Σ\). Show that \(∇· \textbf{E} = 4πρ\). This is one of Maxwell’s Equations .
Solution
By the Divergence Theorem, we have
\[\nonumber \begin{align} \iiint\limits_S ∇· \textbf{E} dV &= \iint\limits_Σ \textbf{E}· dσ \\[4pt] \nonumber &= 4π \iiint\limits_S ρ \,dV \text{ by Gauss’ Law, so combining the integrals gives} \\[4pt] \nonumber \iiint\limits_S (∇· \textbf{E}−4πρ) \,dV &= 0 \text{ , so} \\[4pt] \nonumber ∇· \textbf{E}−4πρ &= 0 \text{ since }Σ \text{ and hence } S \text{ was arbitrary, so} \\[4pt] \nonumber ∇· \textbf{E} &= 4πρ. \\[4pt] \end{align}\]
Often (especially in physics) it is convenient to use other coordinate systems when dealing with quantities such as the gradient, divergence, curl and Laplacian. We will present the formulas for these in cylindrical and spherical coordinates.
Recall from Section 1.7 that a point \((x, y, z)\) can be represented in cylindrical coordinates \((r,θ, z), \text{ where }x = r \cos θ , y = r \sin θ, z = z.\) At each point \((r,θ, z), \text{ let }\textbf{e}_r, \textbf{e}_θ, \textbf{e}_z \text{ be unit vectors in the direction of increasing }r, θ, z,\) respectively (see Figure 4.6.1). Then \(\textbf{e}_r, \textbf{e}_θ, \textbf{e}_z\) form an orthonormal set of vectors. Note, by the right-hand rule, that \(\textbf{e}_z × \textbf{e}_r = \textbf{e}_θ.\)
Similarly, a point \((x, y, z)\) can be represented in spherical coordinates \((ρ,θ,φ)\), where \(x = ρ \sin φ \cos θ, y = ρ \sin φ \sin θ, z = ρ \cos φ.\) At each point \((ρ,θ,φ)\), let \(\textbf{e}_ρ, \textbf{e}_θ, \textbf{e}_φ\) be unit vectors in the direction of increasing \(ρ, θ, φ\), respectively (see Figure 4.6.2). Then the vectors \(\textbf{e}_ρ, \textbf{e}_θ, \textbf{e}_φ\) are orthonormal. By the right-hand rule, we see that \(\textbf{e}_θ × \textbf{e}_ρ = \textbf{e}_φ\).
We can now summarize the expressions for the gradient, divergence, curl and Laplacian in Cartesian, cylindrical and spherical coordinates in the following tables:
Cartesian
\((x, y, z)\): Scalar function \(F\); Vector field \(\textbf{f} = f_1 \textbf{i}+ f_2 \textbf{j}+ f_3\textbf{k}\)
- gradient : \(∇F = \dfrac{∂F}{ ∂x}\textbf{i} + \dfrac{∂F}{ ∂y}\textbf{j} + \dfrac{∂F}{ ∂z}\textbf{k}\)
- divergence : \(∇·\textbf{f} = \dfrac{∂f_1}{ ∂x} + \dfrac{∂f_2}{ ∂y} + \dfrac{∂f_3}{ ∂z}\)
- curl : \(∇ × \textbf{f} = \left ( \dfrac{∂f_3}{ ∂y} − \dfrac{∂f_2}{ ∂z} \right ) \textbf{i} + \left ( \dfrac{∂f_1}{ ∂z} − \dfrac{∂f_3}{ ∂x} \right ) \textbf{j} + \left ( \dfrac{∂f_2}{ ∂x} − \dfrac{∂f_1}{ ∂y} \right )\textbf{k}\)
- Laplacian : \(∆F = \dfrac{∂^ 2F}{ ∂x^ 2} + \dfrac{∂^ 2F}{ ∂y^ 2} + \dfrac{∂^ 2F}{ ∂z^ 2}\)
Cylindrical
\((r,θ, z)\): Scalar function \(F\); Vector field \(\textbf{f} = f_r \textbf{e}_r + f_θ \textbf{e}_θ + f_z \textbf{e}_z\)
- gradient : \(∇F = \dfrac{∂F}{ ∂r} \textbf{e}_r + \dfrac{1}{ r}\dfrac{ ∂F}{ ∂θ} \textbf{e}_θ + \dfrac{∂F}{ ∂z} \textbf{e}_z\)
- divergence : \(∇·\textbf{f} = \dfrac{1}{ r}\dfrac{ ∂}{ ∂r} (r f_r) + \dfrac{1}{ r} \dfrac{∂f_θ}{ ∂θ} + \dfrac{∂f_z}{ ∂z}\)
- curl : \(∇ × \textbf{f} = \left ( \dfrac{1}{ r} \dfrac{∂f_z}{ ∂θ} − \dfrac{∂f_θ}{ ∂z} \right ) \textbf{e}_r + \left ( \dfrac{∂f_r}{ ∂z} − \dfrac{∂f_z}{ ∂r} \right ) \textbf{e}_θ + \dfrac{1}{ r} \left ( \dfrac{∂}{ ∂r} (r f_θ)− \dfrac{∂f_r}{ ∂θ}\right ) \textbf{e}_z\)
- Laplacian : \(∆F = \dfrac{1}{ r}\dfrac{ ∂}{ ∂r} \left ( r \dfrac{∂F}{ ∂r} \right ) + \dfrac{1}{ r^2} \dfrac{∂^ 2F}{ ∂θ^2} + \dfrac{∂^ 2F}{ ∂z^ 2}\)
Spherical
\((ρ,θ,φ)\): Scalar function \(F\); Vector field \(\textbf{f} = f_ρ \textbf{e}_ρ + f_θ \textbf{e}_θ + f_φ \textbf{e}_φ\)
- gradient : \(∇F = \dfrac{∂F}{ ∂ρ} \textbf{e}_ρ + \dfrac{1}{ ρ \sin φ} \dfrac{∂F}{ ∂θ} \textbf{e}_θ + \dfrac{1}{ ρ}\dfrac{ ∂F}{ ∂φ} \textbf{e}_φ\)
- divergence : \(∇·\textbf{f} = \dfrac{1}{ ρ^ 2} \dfrac{∂}{ ∂ρ} (ρ^ 2 f_ρ) + \dfrac{1}{ ρ} \sin φ \dfrac{∂f_θ}{ ∂θ} + \dfrac{1}{ ρ \sin φ} \dfrac{∂}{ ∂φ} (\sin φ f_θ)\)
- curl : \(∇ × \textbf{f} = \dfrac{1}{ ρ \sin φ} \left ( \dfrac{∂}{ ∂φ} (\sin φ f_θ)− \dfrac{∂f_φ}{ ∂θ} \right ) \textbf{e}_ρ + \dfrac{1}{ ρ} \left ( \dfrac{∂}{ ∂ρ} (ρ f_φ)− \dfrac{∂f_ρ}{ ∂φ} \right ) \textbf{e}_θ + \left ( \dfrac{1}{ ρ \sin φ} \dfrac{∂f_ρ}{ ∂θ} − \dfrac{1}{ ρ} \dfrac{∂}{ ∂ρ} (ρ f_θ) \right ) \textbf{e}_φ\)
- Laplacian : \(∆F = \dfrac{1}{ ρ^ 2} \dfrac{∂}{ ∂ρ} \left ( ρ^ 2 \dfrac{∂F}{ ∂ρ} \right ) + \dfrac{1}{ ρ^ 2 \sin^2 φ} \dfrac{∂^ 2F}{ ∂θ^2} + \dfrac{1}{ ρ^ 2 \sin φ} \dfrac{∂}{ ∂φ} \left ( \sin φ \dfrac{∂F}{ ∂φ}\right ) \)
The derivation of the above formulas for cylindrical and spherical coordinates is straightforward but extremely tedious. The basic idea is to take the Cartesian equivalent of the quantity in question and to substitute into that formula using the appropriate coordinate transformation. As an example, we will derive the formula for the gradient in spherical coordinates.
Goal : Show that the gradient of a real-valued function \(F(ρ,θ,φ)\) in spherical coordinates is:
\[\nonumber ∇F = \dfrac{∂F}{ ∂ρ} \textbf{e}_ρ + \dfrac{1}{ ρ \sin φ} \dfrac{∂F}{ ∂θ} \textbf{e}_θ +\dfrac{1}{ ρ}\dfrac{ ∂F}{ ∂φ}\textbf{e}_φ\]
Idea : In the Cartesian gradient formula \(∇F(x, y, z) = \dfrac{∂F}{ ∂x}\textbf{i}+ \dfrac{∂F}{ ∂y}\textbf{j}+ \dfrac{∂F}{ ∂z}\textbf{k}\), put the Cartesian basis vectors i , j , k in terms of the spherical coordinate basis vectors \(\textbf{e}_ρ, \textbf{e}_θ, \textbf{e}_φ\) and functions of \(ρ, θ \text{ and }φ\). Then put the partial derivatives \(\dfrac{∂F}{ ∂x} , \dfrac{∂F}{ ∂y} , \dfrac{∂F}{ ∂z}\) in terms of \(\dfrac{∂F}{ ∂ρ} , \dfrac{∂F}{ ∂θ} , \dfrac{∂F}{ ∂φ}\) and functions of \(ρ, θ \text{ and }φ\).
Step 1 : Get formulas for \(\textbf{e}_ρ, \textbf{e}_θ, \textbf{e}_φ\) in terms of i , j , k .
We can see from Figure 4.6.2 that the unit vector \(\textbf{e}_ρ\) in the \(ρ\) direction at a general point \((ρ,θ,φ)\) is \(\textbf{e}_ρ = \dfrac{\textbf{r}}{ \lVert \textbf{r} \rVert } , \text{ where }\textbf{r} = x\textbf{i} + y\textbf{j} + z\textbf{k}\) is the position vector of the point in Cartesian coordinates. Thus,
\[\nonumber \textbf{e}_ρ = \dfrac{\textbf{r}}{ \lVert \textbf{r} \rVert} = \dfrac{x\textbf{i}+ y\textbf{j}+ z\textbf{k}}{ \sqrt{ x^ 2 + y^ 2 + z^ 2}} ,\]
so using \(x = ρ \sin φ \cos θ, y = ρ \sin φ \sin θ, z = ρ \cos φ, \text{ and }ρ = \sqrt{ x^ 2 + y^ 2 + z^ 2}\), we get:
\[\nonumber \fbox{\(\textbf{e}_ρ = \sin φ \cos θ\textbf{i} + \sin φ \sin θ \textbf{j} + \cos φ\textbf{k}\)}\]
Now, since the angle \(θ\) is measured in the \(x y\)-plane, then the unit vector \(\textbf{e}_θ \text{ in the }θ\) direction must be parallel to the \(x y\)-plane. That is, \(\textbf{e}_θ \text{ is of the form }a\textbf{i} + b\textbf{j} + 0\textbf{k}\). To figure out what \(a \text{ and }b\) are, note that since \(\textbf{e}_θ \perp \textbf{e}_ρ\), then in particular \(\textbf{e}_θ \perp \textbf{e}_ρ\) when \(\textbf{e}_ρ \text{ is in the }x y\)-plane. That occurs when the angle \(φ \text{ is }π/2\). Putting \(φ = π/2\) into the formula for \(\textbf{e}_ρ \text{ gives }\textbf{e}_ρ = \cos θ \textbf{i}+\sin θ \textbf{j}+0\textbf{k}\), and we see that a vector perpendicular to that is \(−\sin θ \textbf{i}+\cos θ \textbf{j}+0\textbf{k}\). Since this vector is also a unit vector and points in the (positive) \(θ\) direction, it must be \(\textbf{e}_θ\):
\[\nonumber \fbox{\(\textbf{e}_θ = −\sin θ \textbf{i} + \cos θ \textbf{j} + 0\textbf{k}\)}\]
Lastly, since \(\textbf{e}_φ = \textbf{e}_θ × \textbf{e}_ρ,\) we get:
\[\nonumber \fbox{\(\textbf{e}_φ = \cos φ \cos θ \textbf{i} + \cos φ \sin θ \textbf{j} − \sin φ\textbf{k}\)}\]
Step 2 : Use the three formulas from Step 1 to solve for i , j , k in terms of \(\textbf{e}_ρ, \textbf{e}_θ, \textbf{e}_φ\).
This comes down to solving a system of three equations in three unknowns. There are many ways of doing this, but we will do it by combining the formulas for \(\textbf{e}_ρ \text{ and }\textbf{e}_φ \text{ to eliminate }k\), which will give us an equation involving just i and j . This, with the formula for \(\textbf{e}_θ\), will then leave us with a system of two equations in two unknowns ( i and j ), which we will use to solve first for j then for i . Lastly, we will solve for k .
First, note that
\[\nonumber \sin φ \textbf{e}_ρ + \cos φ \textbf{e}_φ = \cos θ \textbf{i} + \sin θ \textbf{j}\]
so that
\[\nonumber \sin θ (\sin φ \textbf{e}_ρ + \cos φ \textbf{e}_φ) + \cos θ \textbf{e}_θ = (\sin^2 θ +\cos^2 θ)\textbf{j} = \textbf{j} ,\]
and so:
\[\nonumber \fbox{\(\textbf{j} = \sin φ \sin θ \textbf{e}_ρ + \cos θ \textbf{e}_θ + \cos φ \sin θ \textbf{e}_φ\)}\]
Likewise, we see that
\[\nonumber \cos θ (\sin φ \textbf{e}_ρ + \cos φ \textbf{e}_φ) − \sin θ \textbf{e}_θ = (\cos^2 θ +\sin^2 θ)\textbf{i} = \textbf{i} ,\]
and so:
\[\nonumber \fbox{\(\textbf{i} = \sin φ \cos θ \textbf{e}_ρ − \sin θ \textbf{e}_θ + \cos φ \cos θ \textbf{e}_φ\)}\]
Lastly, we see that:
\[\nonumber \fbox{\(\textbf{k} = \cos φ \textbf{e}_ρ − \sin φ \textbf{e}_φ\)}\]
Step 3: Get formulas for \(\dfrac{∂F}{ ∂ρ} , \dfrac{∂F}{ ∂θ} , \dfrac{∂F}{ ∂φ} \text{ in terms of }\dfrac{∂F}{ ∂x} , \dfrac{∂F}{ ∂y} , \dfrac{∂F}{ ∂z}\).
By the Chain Rule, we have
\[\nonumber \begin{align} \dfrac{∂F}{ ∂ρ} &= \dfrac{∂F}{ ∂x} \dfrac{∂x}{ ∂ρ} + \dfrac{∂F}{ ∂y}\dfrac{ ∂y}{ ∂ρ} + \dfrac{∂F}{ ∂z} \dfrac{∂z}{ ∂ρ}, \\[4pt] \nonumber \dfrac{∂F}{ ∂θ} &= \dfrac{∂F}{ ∂x} \dfrac{∂x}{ ∂θ} + \dfrac{∂F}{ ∂y} \dfrac{∂y}{ ∂θ} + \dfrac{∂F}{ ∂z} \dfrac{∂z}{ ∂θ}, \\[4pt] \nonumber \dfrac{∂F}{ ∂φ} &= \dfrac{∂F}{ ∂x} \dfrac{∂x}{ ∂φ} + \dfrac{∂F}{ ∂y} \dfrac{∂y}{ ∂φ} + \dfrac{∂F}{ ∂z} \dfrac{∂z}{ ∂φ} ,\\[4pt] \end{align}\]
which yields:
\[\fbox{\(\nonumber \begin{align} \dfrac{∂F}{ ∂ρ} &= \sin φ \cos θ\dfrac{∂F}{ ∂x} + \sin φ \sin θ \dfrac{∂F}{ ∂y} + \cos φ \dfrac{∂F}{ ∂z} \\[4pt] \nonumber \dfrac{∂F}{ ∂θ} &= −ρ \sin φ \sin θ \dfrac{∂F}{ ∂x} + ρ \sin φ \cos θ \dfrac{∂F}{ ∂y} \\[4pt] \nonumber \dfrac{∂F}{ ∂φ} &= ρ \cos φ \cos θ \dfrac{∂F}{ ∂x} + ρ \cos φ \sin θ \dfrac{∂F}{ ∂y} − ρ \sin φ \dfrac{∂F}{ ∂z} \\[4pt] \end{align}\)}\]
Step 4 : Use the three formulas from Step 3 to solve for \(\dfrac{∂F}{ ∂x} , \dfrac{∂F}{ ∂y} , \dfrac{∂F}{ ∂z}\) in terms of \(\dfrac{∂F}{ ∂ρ} , \dfrac{∂F}{ ∂θ} , \dfrac{∂F}{ ∂φ} \).
Again, this involves solving a system of three equations in three unknowns. Using a similar process of elimination as in Step 2, we get:
\[\fbox{\(\nonumber \begin{align}\dfrac{∂F}{ ∂x} &= \dfrac{1}{ ρ \sin φ} \left ( ρ \sin^2 φ \cos θ \dfrac{∂F}{ ∂ρ} − \sin θ \dfrac{∂F}{ ∂θ} + \sin φ \cos φ \cos θ \dfrac{∂F}{ ∂φ}\right ) \\[4pt] \nonumber \dfrac{∂F}{ ∂y} &= \dfrac{1}{ ρ \sin φ} \left ( ρ \sin^2 φ \sin θ \dfrac{∂F}{ ∂ρ} + \cos θ \dfrac{∂F}{ ∂θ} + \sin φ \cos φ \sin θ \dfrac{∂F}{ ∂φ}\right ) \\[4pt] \nonumber \dfrac{∂F}{ ∂z} &= \dfrac{1}{ρ} \left ( ρ \cos φ \dfrac{∂F}{ ∂ρ} − \sin φ \dfrac{∂F}{ ∂φ}\right ) \\[4pt] \end{align}\)}\]
Step 5: Substitute the formulas for i , j , k from Step 2 and the formulas for \(\dfrac{∂F}{ ∂x} , \dfrac{∂F}{ ∂y} , \dfrac{∂F}{ ∂z}\) from Step 4 into the Cartesian gradient formula \(∇F(x, y, z) = \dfrac{∂F}{ ∂x} \textbf{i}+ \dfrac{∂F}{ ∂y} \textbf{j}+ \dfrac{∂F}{ ∂z} \textbf{k}\).
Doing this last step is perhaps the most tedious, since it involves simplifying \(3×3+3×3+ 2×2 = 22\) terms! Namely,
\[\nonumber \begin{align} ∇F &= \dfrac{1}{ ρ \sin φ} \left ( ρ \sin^2 φ \cos θ \dfrac{∂F}{ ∂ρ} −\sin θ \dfrac{∂F}{ ∂θ} +\sin φ \cos φ \cos θ \dfrac{∂F}{ ∂φ}\right ) (\sin φ \cos θ \textbf{e}_ρ −\sin θ \textbf{e}_θ +\cos φ \cos θ \textbf{e}_φ) \\[4pt] \nonumber &+\dfrac{1}{ ρ \sin φ} \left ( ρ \sin^2 φ \sin θ \dfrac{∂F}{ ∂ρ} +\cos θ \dfrac{∂F}{ ∂θ} +\sin φ \cos φ \sin θ \dfrac{∂F}{ ∂φ}\right ) (\sin φ \sin θ \textbf{e}_ρ +\cos θ \textbf{e}_θ +\cos φ \sin θ \textbf{e}_φ)\\[4pt] \nonumber &+\dfrac{1}{ ρ} \left ( ρ \cos φ \dfrac{∂F}{ ∂ρ} −\sin φ \dfrac{∂F}{ ∂φ}\right ) (\cos φ \textbf{e}_ρ −\sin φ \textbf{e}_φ) ,\\[4pt] \end{align}\]
which we see has 8 terms involving \(\textbf{e}_ρ\), 6 terms involving \(\textbf{e}_θ\), and 8 terms involving \(\textbf{e}_φ\). But the algebra is straightforward and yields the desired result:
\[∇F = \dfrac{∂F}{ ∂ρ} \textbf{e}_ρ + \dfrac{1}{ ρ \sin φ} \dfrac{∂F}{ ∂θ} \textbf{e}_θ + \dfrac{1}{ ρ} \dfrac{∂F}{ ∂φ} \textbf{e}_φ \quad \checkmark \]
Example 4.19
In Example 4.17 we showed that \(∇\lVert \textbf{r} \rVert ^2 = 2\textbf{r} \text{ and }∆\lVert \textbf{r} \rVert ^2 = 6, \text{ where }\textbf{r}(x, y, z) = x\textbf{i} + y\textbf{j} + z\textbf{k}\) in Cartesian coordinates. Verify that we get the same answers if we switch to spherical coordinates.
Solution
Since \(\lVert \textbf{r} \rVert ^2 = x^ 2 + y^ 2 + z^ 2 = ρ^ 2 \text{ in spherical coordinates, let }F(ρ,θ,φ) = ρ^ 2 \) (so that \(F(ρ,θ,φ) = \lVert \textbf{r} \rVert ^2\) ). The gradient of \(F\) in spherical coordinates is
\[\nonumber \begin{align} ∇F &= \dfrac{∂F}{ ∂ρ} \textbf{e}_ρ + \dfrac{1}{ ρ \sin φ} \dfrac{∂F}{ ∂θ} \textbf{e}_θ + \dfrac{1}{ ρ} \dfrac{∂F}{ ∂φ} \textbf{e}_φ \\[4pt] \nonumber &=2ρ \textbf{e}_ρ + \dfrac{1}{ ρ \sin φ} (0)\textbf{e}_θ + \dfrac{1}{ ρ} (0)\textbf{e}_φ \\[4pt] \nonumber &= 2ρ \textbf{e}_ρ = 2ρ \dfrac{\textbf{r}}{ \lVert \textbf{r} \rVert} , \text{ as we showed earlier, so}\\[4pt] \nonumber &= 2ρ \dfrac{\textbf{r}}{ ρ} = 2\textbf{r} , \text{ as expected. And the Laplacian is} \\[4pt] \nonumber ∆F &= \dfrac{1}{ ρ^ 2} \dfrac{∂}{ ∂ρ} \left ( ρ^2 \dfrac{∂F}{ ∂ρ} \right ) + \dfrac{1}{ ρ^ 2 \sin^2 φ} \dfrac{∂^2F}{ ∂θ^2} + \dfrac{1}{ ρ^ 2 \sin φ} \dfrac{∂}{ ∂φ} \left ( \sin φ \dfrac{∂F}{ ∂φ}\right ) \\[4pt] \nonumber &= \dfrac{1}{ ρ^ 2} \dfrac{∂}{ ∂ρ} (ρ^ 2 2ρ) + \dfrac{1}{ ρ^ 2 \sin φ} (0) + \dfrac{1}{ ρ^ 2 \sin φ} \dfrac{∂}{ ∂φ} \left ( \sin φ(0)\right ) \\[4pt] \nonumber &= \dfrac{1}{ ρ^ 2} \dfrac{∂}{ ∂ρ} (2ρ^ 3 ) + 0 + 0 \\[4pt] \nonumber &= \dfrac{1}{ ρ^ 2} (6ρ^ 2 ) = 6 , \text{ as expected.} \\[4pt] \end{align}\]


9.3: Intersectionality
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- Erika Gutierrez, Janét Hund, Shaheen Johnson, Carlos Ramos, Lisette Rodriguez, & Joy Tsuhako
- Long Beach City College, Cerritos College, & Saddleback College via ASCCC Open Educational Resources Initiative (OERI)
Women and Gender Issues
Where Do Women Fit In?
Asian America has masked a series of internal tensions. In order to produce a sense of racial solidarity, Asian American activists framed social injustices in terms of race, veiling other competing social categories such as gender, sexuality, ethnicity, and nationality. The relative absence of gender as a lens for Asian American activism and resistance throughout the 1970s until the present should therefore be read as neither an indication of the absence of gender inequality nor of the disengagement of Asian American women from issues of social justice.
Many Asian American activists (including some of the authors in this book) refute the label "feminist" although their work pays special attention to the experiences of women. Sometimes this feeling reflects a fear of alienating men -- a consequence that seems inevitable if men are unable to own up to their gender privilege. At other times, the antipathy towards feminism reflects the cultural insensitivity and racism of white, European feminists.
Dragon Ladies: A Brief History
Empress Tsu-his ruled China from 1898 to 1908 from the Dragon Throne. The New York Times described her as "the wicked witch of the East, a reptilian dragon lady who had arranged the poisoning, strangling, beheading, or forced suicide of anyone who had ever challenged her autocratic rule." The shadow of the Dragon Lady -- with her cruel, perverse, and inhuman ways -- continued to darken encounters between Asian women and the West they flocked to for refuge.
Far from being predatory, many of the first Asian women to come to the U.S. in the mid-1800s were disadvantaged Chinese women, who were tricked, kidnapped, or smuggled into the country to serve the predominantly male Chinese community as prostitutes. The impression that all Asian women were prostitutes, born at that time, "colored the public perception of, attitude toward, and action against all Chinese women for almost a century," writes historian Sucheng Chan.
Police and legislators singled out Chinese women for special restrictions "not so much because they were prostitutes as such (since there were also many white prostitutes around) but because -- as Chinese -- they allegedly brought in especially virulent strains of venereal diseases, introduced opium addiction, and enticed white boys into a life of sin," Chan also writes. Chinese women who were not prostitutes ended up bearing the brunt of the Chinese exclusion laws that passed in the late 1800s.
During these years, Japanese immigration stepped up, and with it, a reactionary anti-Japanese movement joined established anti-Chinese sentiment. During the early 1900s, Japanese numbered less than 3 percent of the total population in California, but nevertheless encountered virulent and sometimes violent racism. The "picture brides" from Japan who emigrated to join their husbands in the U.S. were, to racist Californians, "another example of Oriental treachery," according to historian Roger Daniels.
It bears noting that despite the fact that they weren't in the country in large numbers, Asian women shouldered much of the cost of subsidizing Asian men's labor. U.S. employers didn't have to pay Asian men as much as other laborers who had families to support, since Asian women in Asian bore the costs of rearing children and taking care of the older generation.
Asian women who did emigrate here before the 1960s were also usually employed as cheap labor. In the pre-World War II years, close to half of all Japanese American women were employed as servants or laundresses in the San Francisco area. The World War II internment of Japanese Americans made them especially easy to exploit: they had lost their homes, possessions, and savings when forcibly interned at the camps, Yet, in order to leave, they had to prove they had jobs and homes. U.S. government officials thoughtfully arranged for their employment by fielding requests, most of which were for servants.
Immigration Characteristics
Issues concerning immigration affect many aspects of the Asian American community. This is understandable since almost two-thirds of all Asian Americans are foreign-born. Before trying to examine the many controversies regarding the benefits or costs of immigration, we first need to examine the characteristics of the immigrant population, Asian and otherwise.
The Immigrant and U.S. Born Populations
The data in the following table was calculated using the 2000 Census 1% Public Use Microdata Samples, and they compare different immigrant groups (based on their home country area) with each other and with all those who are either U.S.-born or foreign-born in the U.S. on different measures of socioeconomic achievement. You can click on a column heading to sort up or down. You can also read the detailed description of the methodology and terminology used to create the statistics.
The statistics include immigrants from all countries, not just those from Asia. According to the 2000 Census, the immigrant/foreign-born population of the U.S. was just about 28,910,800. Of these, 5.5% were Black, 25.9% were Asian, 46.4% were Hispanic/Latino, and 22.1% were white. The statistics below represent sound research but different statisticis can be used to support both sides of an issue. So you can choose to agree with my conclusions or not.
We should first understand that immigrating to another country is not an easy thing to do. It almost always involves making elaborate preparations and costs a lot of money. Many times it also means giving up personal relationships at home (at least temporarily, if not permanently) and learning a new language and culture. The point is, not everyone who wants to immigrate actually does. In fact, those who are very poor rarely immigrate -- they just don't have the resources. Those who do immigrate tend to be from their country's middle and professional classes.
This point is illustrated by the results from the table, which compares various socioeconomic characteristics between U.S.-born and immigrant groups by their home country area. To view the full-size table of statistics, click on Table 9.3.2. Once the table appears, you can click on a column heading to sort up or down. You can also read the detailed description of the methodology and terminology used to create the statistics.
The results show that immigrants as a group actually have a slightly higher college degree attainment rate and a much higher rate of having an advanced degree (medical, law, or doctorate) than do the U.S.-born. On both measures, immigrants from Africa actually have the highest educational achievement rates and they also have the lowest rate of having less than a high school education. African immigrants are also most likely to be in the labor market.
Therefore, it's clear that immigrants from Africa tend to come from their country's elite classes. In contrast, the statistics point out that immigrants from Latin and South America and from the Caribbean have the lowest educational attainment rates. We can probably surmise from this that they are more likely to be from rural or working class backgrounds. As another example of this implication, immigrants from Latin/South America and the Caribbean have the lowest median personal (per capita) income, as well as the highest rates of living in poverty and receiving public assistance.
In addition, they have the lowest rates of being married with spouse present, working in a high skill (executive, professional, technical, or upper management) occupation and the lowest media socioeconomic index (SEI) score, a measure of occupational prestige. However, these statistics do not necessarily lead to the conclusion that immigrants from Latin/South America and the Caribbean are a drain on the U.S. economy or that they consume more benefits than they contribute. For a discussion of that issue, be sure to read the article on the impacts of immigration .
Other Groups and Their Levels of Success
In regard to other immigrant groups, the statistics above show that immigrants from Asia and Pacific Islands compare quite favorably to other immigrants and to the U.S.-born as well. However, there also seems to be a much wider spread of characteristics among Asian immigrants. In other words, there seems to be many who are more likely to be from rural or working class backgrounds (and therefore have lower socioeconomic attainment rates), along with many other Asian immigrants from middle class and professional backgrounds who have very high attainment rates.
For example, Asian and Pacific Islander immigrants have a much rate of not being proficient in English than do the U.S.-born (which is understandable since English is a foreign language to most Asians) and they also have a higher rate of less than high school completion than do the U.S.-born. On the other hand, Asian & Pacific Islander immigrants have a median personal (per capita) income comparable to the U.S.-born, along with a much higher median family income. They also have higher rates of having a college degree, an advanced degree, and working at a high skill occupation than do the U.S.-born.
Similarly, immigrants from Europe, Russia, and Canada tend to have socioeconomic attainment levels that are very comparable to that for the U.S.-born and in several categories, outperform them as well. These include higher rates of having a college degree, an advanced degree, working in a high skill occupation, and most notably, the highest median personal (per capita) income of all groups in the table. Interestingly, they also have the lowest rate of being in the labor market, which may suggest that many are retired but rather affluent as well.
Overall, all of these socioeconomic measures and statistics comparing immigrants to the U.S.-born population suggest that in most cases, both groups are relatively close to the other. But again, these numbers can be used to support both sides of the immigration debate -- that immigrants are not achieving as well as the U.S.-born and vice-versa. However, it does seem clear that these statistics do not support the stereotype of immigrants as being chronically unemployed, in poverty, and on public assistance. They do suggest that just like any other social group in the U.S., there is a lot of diversity within each group and that we as a society should be careful about making sweeping generalizations about all members of a particular group.
Religion, Spirituality & Faith
Among the more traditional elements of Asian American culture, religion, spirituality, and faith have always been important to Asian American communities, as they were for many generations before them. But within the diversity of the Asian American community, so too comes diversity in our religious beliefs and practices.
Which Religion is the Most Popular?
One of the first questions to examine is, which religions or faith traditions are the most popular among Asian Americans and among each of the different Asian ethnic groups? Unfortunately, nationally representative and reliable statistics are difficult to find. There are few studies or data that would answer these questions conclusively, particularly ones that break down religious affiliation among different Asian ethnic groups.
|
American Religious Identfication Survey 1990-2008: Asian Americans |
|||||
|---|---|---|---|---|---|
| 1990 | 2001 | 2008 | |||
| None/Agnostic | 16% | 22% | 27% | ||
| Eastern Religions | 8% | 22% | 21% | ||
| Catholic | 27% | 20% | 17% | ||
| Other Christian | |||||
| 13% | 11% | 10% | |||
| 11% | 6% | 6% | |||
| 9% | 4% | 3% | |||
| 3% | 2% | 2% | |||
|
Mormon |
2% | 0% | 0% | ||
| Muslim | 3% | 8% | 8% | ||
| New Religious Movements | 2% | 1% | 2% | ||
| Jewish | 1% | 0% | 0% | ||
| Don't Know/ Declined to Answer | 4% | 5% | 5% |
Nonetheless, there are some statistics that give a general picture of religious affiliation within the Asian American community. One of the largest, most up to date, and most comprehensive sources is the American Religious Identification Survey (ARIS), conducted by researchers at Trinity College (CT). The ARIS was first conducted in 1990, again in 2000, and the most recent wave was completed in 2008. The 2008 study includes data from a large, nationally representative sample of 54,461 U.S. adults in the 48 contiguous states.
The following Table 9.3.5 is taken from the ARIS 2008 report . The results show that while no religion can claim a majority of followers in the Asian American community, as of 2008, those who claim no religious affiliation are the largest group. In fact, this group has grown significantly since the first ARIS study in 1990 and its percentage in 2008 (27%) among Asian American is the largest of all the major racial ethnic groups in the study (whites are second with 16% claiming no religious affiliation). The second-largest religious group among Asian Americans are "Eastern Religions" that include Buddhist, Hindu, Taoist, Baha'i, Shintoist, Zoroastrian, and Sikh. These Eastern Religions saw a dramatic increase from 1990 to 2001, then leveled off in 2008. Catholics are the third-largest group at 17% in 2008, with their proportions declining notably from 27% in 1990.
| 2008 | |||
| Christian | 45% | ||
| Protestant | 27% | ||
| Evangelical | 17% | ||
| Mainline | 9% | ||
| Historically Black | < 0.5% | ||
| Catholic | 17% | ||
| Mormon | 1% | ||
| Jehovah's Witness | < 0.5% | ||
| Orthodox | < 0.5% | ||
| Other Christian | < 0.5% | ||
| Eastern & Other Religions | 30% | ||
| Hindu | 14% | ||
| Buddhist | 9% | ||
| Muslim | 4% | ||
| Other World Religions | 2% | ||
| Other Faiths | 1% | ||
| Jewish | < 0.5% | ||
| Unaffiliated | 23% | ||
| Secular Unaffiliated | 11% | ||
| Religious Unaffiliated | 5% | ||
| Agnostic | 4% | ||
| Atheist | 3% | ||
| Don't Know/Refused | 2% |
The category of "Christian Generic" (comprising those who identified as Christian, Protestant, Evangelical/ Born Again Christian, Born Again, Fundamentalist, Independent Christian, Missionary Alliance Church, and Non-Denominational Christian) is the fourth-largest group at 10% in 2008. Other Christian and Protestant denominations are listed below that. The results show that in 2008, Muslims represented 8% of the Asian American population (up from 3% in 1990) and "New Religious Movements" (comprising those who identified as Scientology, New Age, Eckankar, Spiritualist, Unitarian-Universalist, Deist, Wiccan, Pagan, Druid, Indian Religion, Santeria, and Rastafarian) claiming 2% in 2008.
These results are largely confirmed by a second comprehensive survey of religious identification taken in 2008, the U.S. Religious Landscape Survey (1.2 MB), a national survey of over 35,000 respondents conducted by the Pew Forum on Religion and Public Life .
In contrast to the ARIS 2008 report, the USLRS methodology sometimes includes the same denomination with separate categories (i.e., Baptists can be both "Evangelical" and "Mainline") -- please check page 12 and Appendix 2 of the USLRS report for the exact categorizations and their detailed explanation of their methodology. The data shown here is for Asian American respondents only and is taken from page 40 of their report.
Again the data show that Christian faiths and denominations claim the highest percentage of followers among Asian Americans, with Eastern Religions and unaffiliated responses also claiming large numbers of respondents. Interesting, once the unique faiths within the "Eastern Religions" category are expanded, we see that Hinduism is the mos popular eastern faith among Asian Americans (due largely to the large size of the Indian American population), with Buddhism second.
Unfortunately, neither the ARIS nor the USLRS studies break the religious affiliation down to specific Asian ethnic groups. For that matter, I have yet to find any research that does. So to try to measure the size of religions within each ethnic group, we can look at the proportions for different religions within that Asian country. Although it's not completely accurate, it's a generally safe assumption that the religious proportions within an Asian country are similar to that within its community in the U.S., since the majority of Asian Americans are foreign-born, as stated in the 2000 CIA World Factbook :
- Bangladesh: Muslim 88.3%, Hindu 10.5%, other 1.2%
- India: Hindu 80%, Muslim 14%, Christian 2.4%, Sikh 2%, Buddhist 0.7%, Jains 0.5%, other 0.4%
- Philippines: Roman Catholic 83%, Protestant 9%, Muslim 5%, Buddhist and other 3%
- Japan: observe both Shinto and Buddhist 84%, other 16% (including Christian 0.7%)
- South Korea: Christian 49%, Buddhist 47%, Confucianist 3%, Shamanist, Chondogyo (Religion of the Heavenly Way), and other 1%
Again, these stats are imperfect because as China and Viet Nam are both officially atheist countries, there are no statistics on the proportions of religions in each country.
How Religion, Spirituality, and Faith Help
Ultimately, as there is so much diversity in the Asian American population in so many ways, so too this applies to our religions and practices of spirituality and faith. But they all share the commonality of helping Asian Americans adjust to life in the U.S. and all the issues that surround what it means to be an Asian American.
As several social scientists point out, these various forms of spirituality and faith help Asian Americans to deal with the upheavals of immigration, adapting to a new country, and other difficult personal and social transformations by providing a safe and comfortable environment in which immigrants can socialize, share information, and assist each other. In this process, religious traditions can help in the process of forming Asian immigrant communities by giving specific Asian ethnic groups another source of solidarity, in addition to their common ethnicity, on which to build relationships and cooperation. In fact, history shows that numerous churches and religious organizations played very important roles in helping immigrants from China, Japan, the Philippines, South Asia, and Korea adjust to life in the U.S.
Also, the secular functions of religion are just as, if not even more important in helping Asian Americans in their everyday lives. Specifically, many churches, temples, and other religious organizations provide their members with important and useful services around practical, everyday matters such as translation assistance. Other practical examples include information and assistance on issues relating to education, employment, housing, health care, business and financial advice, legal advice, marriage counseling, and dealing with their Americanized children, etc. As such, many churches are almost like social service agencies in terms of the ways in which they help Asian Americans in practical, day-to-day matters.
Other scholars and studies show that churches can also provide social status and prestige for their members. As one example sociologist Pyong Gap Min describes that since many Korean immigrants face underemployment due to their lack of English fluency once they immigrate to the U.S. (especially if they come from educated and professional backgrounds in Korea), they often feel ashamed, embarrassed, or alienated as they adjust to their lower status level in the U.S. Within their church however, many Korean immigrants find a sense of status through official positions inside the church. These can include being assistant ministers, education directors, unordained associate pastors, elders, deacons, and committee chairs, etc.
Finally, as Bankston and Zhou point out in their study of the New Orleans Vietnamese community, religion can play a significant part in affecting a young Asian American's ethnic identity. The Catholic churches in the Vietnamese section of the city helped to keep young Vietnamese Americans integrated within the larger community. Those youngsters who attended church and participated in religious activities more were more likely to do well in school and to stay out of trouble .
Of course, religion, spirituality, and faith is only one part of this adaptation and socialization process and it interacts with many other factors in affecting how an Asian immigrant adjusts to his/her new life in the U.S. Nonetheless, its power is undeniable. For hundreds of generations in the past, it has bonded communities and been the basis for many people's lives. Even with changes in culture, physical location, and social institutions, its effect lives on.
 The following is an article entitled "Young, Gay, and APA," originally published in the July 17, 1999 issue of AsianWeek Magazine, written by Joyce Nishioka. It captures many of the obstacles and challenges that LGBT Asian Americans go through as they search for acceptance and happiness with the multiple forms of their personal identities.
Double Jeopardy
Nineteen-year-old Eric Aquino remembers a day not that long ago when he kneeled down to tie his shoe during P.E. class. He looked up to find a boy towering over him, saying, "That's where you belong" and making a comment about oral sex.  "What could I do but ignore it? One thing I always did was ignore it."
While feelings of rejection and questions about "being normal" haunt most adolescents, they often hit harder at those who are minorities, either racial or sexual. And too often, those are the kids who get the least support. A 1989 study from the Department of Health and Human Services found that a gay teen who comes out to his or her parents faced about a 50-50 chance of being rejected and 1 in 4 had to leave home. Ten years later, a study in The Archives of Pediatric and Adolescent Medicine found that gay and bisexual teens are more than three times as likely to attempt suicide as other youths.
Surveys indicate that 80 percent of gay students do not feel safe in schools, and one poll by the Centers for Disease Control and Prevention showed 1 in 13 high school students had been attacked or harassed because they were perceived to be homosexual. Nationwide, 18 percent of all gay students are physically injured to the point they require medical treatment, and they are seven times as likely as their straight peers to be threatened with a weapon at school, according to the Gay, Lesbian and Straight Education Network.
Protecting homosexual Asian teens from discrimination requires double-duty measures, advocates say. Ofie Virtucio, a coordinator for AQUA, San Francisco's only citywide organization for gay Asian American teenagers (now known as the API Wellness Center), maintains that they are especially likely to be closeted and ignored. "Asians are the model minorities," she says, describing a common stereotype.   And there is nothing to adequately protect them."
 "With the anti-Asian sentiment, students are harassed more for being Asian because it's more visible than sexuality." says San Francisco school district counselor Crystal Jang.
The Closet is a Lonely Place to Live
"People don't think there are API gays and lesbians," Virtucio says. "There is hardly any research, and no money goes to them."  But if the often quoted figure of 10 percent of a population holds, the figure could exceed 1,300 in the public junior high and high schools alone. Asian American students, says Jang, account for about 90 percent of the kids she sees through the district's Support Services for Sexual Minorities Youth Program. Though there are more support groups for gay youths than ever before, Virtucio said many Asian American teens find it difficult to fit in. Nor do they have any role models. This decade's most noted gays and lesbians -- actresses Ellen DeGeneres and Anne Heche, Ambassador James Hormel and former Wisconsin congressman Steve Gunderson, Migden and Kuehl -- are all white, and so is society's perception of gay America.
 "How can they be understood when they talk about their close-knit family they can never come out to? They need to see people like them. Even if it's just serving rice, they need something familiar so they could [relate] and feel like they could be part of this community," says Virtucio, who touts her four-year-old group as "a channel to come out." In the summer, 20 to 30 teens -- half of whom are immigrants -- go to AQUA's weekly drop-in sessions. Though the group initially attracted mostly college-age men, most of its members today are younger, and half are female. At a recent get-together, the girls seemed much less vocal than boys, and though several young men agreed to be interviewed, no girls did. Jang explains that girls are more likely than boys to refrain from expressing their sexuality, possibly because of the shame they think they may bring on themselves and their families. One girl, she recalled, fell in love with her godsister and wanted to tell her, but she was afraid that if she did, everyone in Chinatown would find out.
For both genders, though, coming out to family and friends is a huge issue, one that Virtucio says cannot be put off indefinitely. "Parents want to know," she said, adding that many AQUA members have told her that they suspected that their parents knew about their sexuality long before their children would admit it to themselves. Mothers, she said, might ask daughters questions like, "Why to you dress that way? Wear a skirt." Or they might tell their sons, "Don't walk like that." At the same time, she said, cultural pressures to put the family first or to hide one's feelings often convince Asian and Asian American youth to internalize their sexuality. Each family member often is expected to fill an explicit role. For example, she explained, a Filipina, particularly the first-born daughter, "is supposed to take care of the family, and get married and have kids."  He is supposed to extend the family name."
Desmond Kwok says his parents accept his sexual orientation -- though they don't necessarily support him emotionally. He acknowledges an ongoing "starvation for love" that he blames on his parents. Both have been distant, he says, especially his father, a businessman who lives in Chicago. Kwok says he found support for coming out not from his family, but from a gang he was in two years ago. "They were really cool with it, and it boosted my confidence in the whole coming-out process," he said. "They'd say, 'If someone has a grudge against you for being gay, we're there for you. We'll kick their asses.' "
Now, Kwok dates "older" Asian and Asian American men -- at least 19 -- because few come out before then, he says. He admits that he has tried to find boyfriends over the Internet, at bars and cafes, "the worst places to meet a good boyfriend. A graduate of the School of the Arts, a magnet academy, Kwok said he intends to continue his work as an advocate for gay Asian and Asian American teens. Yet even now he cannot rid "the feeling of being alone -- being around people who really love you, but still knowing they are heterosexual. They'll be with their girlfriends or boyfriends, and here I am all alone, sitting around, boo-hoo, no boyfriend."
'Straight' Into Isolation, 'Out' Into Happiness
Eric Aquino never had such peer support growing up in Vallejo, Calif., and especially in junior high school. "I felt alone," Aquino said. He avoided his locker, where the popular kids hung out, and instead took long, circuitous paths to classes to dodge their cruel comments. "A good day for me was being able to walk down the hall without having anyone ask, 'Are you gay?  His grades fell. "I would be late to class and wouldn't bring my books," he explained. "I couldn't concentrate. I looked at the clock until it was 3 o'clock and time to go."
Aquino's high school years were both the happiest and one of the most depressing times of his life. He joined marching band and had friends for the first time, but he also started feeling that he was, in fact, gay. "Friends were important to me because I never had any, but they didn't know me for what I was," he said. Aquino thought perhaps he should wait until he was 18 to come out, so that if his parents rejected him, he could run away. He also considered living in the closet and spent much of his time thinking of ways to keep his secret. "I thought of different alternatives, other options. Like, I'll get married and have kids, [then divorce] and be a single parent, and my parents would just think I never found love again."
Thinking Sociologically
Once LGBTQ Asian Americans come out of the closet, do they find more support and acceptance within the mainstream LGBTQ community? Many do, but unfortunately, anti-Asian racism among the predominantly white LGBTQ community still exists. Joseph Erbentraut's article " Gay Anti-Asian Prejudice Thrives On the Internet " and Gay.net's article "Gay Racism Comes Out" provide insight into the challenges that LGBTQ Asian Americans face with regards to acceptance in the larger LGBTQ community. How are LGBTQ Asian Americans treated in the LGBTQ community in your city?
Ofiee Virtucio, 21, can relate to the feeling of isolation. "Maybe it's the feeling where you know you're Asian but sometimes in situations you're embarrassed to be," she said. "That's where I was for a long time. Of course I was lonely."  She looked me in the eyes; she was worried," Virtucio said. "I said, 'No!'  I still love you, Ofie.' " Two years later, the family came to the United States. "I had to be white in a month," she recalled. "When I started talking, I had an American accent that I could use, so I could make friends," she said.  Most importantly, I had to get friends. I had to get to know what America is all about. I had to survive."
She recalled: "I was trying to be straight but didn't want to have sex.  Though she had a boyfriend in high school, she secretly had crushes on girls, especially the teenage lesbians who were "out." At the same time, she recalls, she "couldn't relate. They were more 'we're-here-we're-queer' ... I knew I was gay, but I thought, 'I'm not like that.' It made me think I could never be like that." So, she said, "When my friends would talk about cute guys, I would jump into the conversation. I thought, 'OK, I have to do this right now,' so I'd say things like, 'Oh, he's so cute.' "Then when I would go home, I'd be like ... oh," said Virtucio, covering her eyes with her palms. "It hurts. It really, really hurts."
  "I knew it was going to be a hard life," she said. "I thought, 'How am I going to tell my siblings? How am I going to get a job?  What are people going to think of me? I thought people would know now -- just because I know I'm gay -- that they'll just see it."
Virtucio never had the opportunity to come out to her mother, who passed away when she was 15. But in college, she did tell her father. She remembers he was in the garden watering plants when he asked her, out of the blue, whether her girlfriend was more than a friend. Startled, Virtucio says she denied it, but later that day, she opened the door to his bedroom and said it was true. They took a walk on the beach after that. "He told me whatever made me happy was fine," Virtucio recalls. "My father used to be mean to my mom, pot-bellied, chauvinistic," she says. "But for some reason he found it in his heart to understand. That moment was amazing for me. I thought if my dad could understand, I really don't care what the world thinks. I'm just going to be the person I am."
Contributors and Attributions
- Tsuhako, Joy. (Cerritos College)
- Gutierrez, Erika. (Santiago Canyon College)
- Asian Nation (Le) ( CC BY-NC-ND ) adapted with permission
Works Cited & Recommended for Further Reading
- Carnes, T. & Yang, F. (Eds.). (2004). Asian American Religions: The Making and Remaking of Borders and Boundaries . New York, NY: New York University Press.
- Cho, S. (1997). Rice: Explorations into Asian Gay Culture & Politics . San Francisco, CA: Queer Press.
- Chou, R.S. (2012). Asian American Sexual Politics: The Construction of Race, Gender, and Sexuality . New York, NY: Rowman & Littlefield Publishers.
- Duncan, P. & Wong. G. (Eds). (2014). Mothering in East Asian Communities . Bradford, ON: Demeter Press.
- Eng, D.L. & Hom, A.Y. (Eds.). (1998). Q & A: Queer in Asian America. Philadelphia, PA: Temple University Press.
- Forbes, B.D., Mahan, J.H. (Eds.). (2017). Religion and Popular Culture in America . Berkeley, CA: University of California Press.
- Fujiwara, L. & Roshanravan S. (Eds.). (2018). Asian American Feminisms and Women of Color Politics . Seattle, WA: University of Washington Press.
- Hune, S. (Ed.). (2020). Our Voices, Our Histories: Asian American and Pacific Islander Women . New York, NY: NYU Press.
- Kang, M. (2010). The Managed Hand: Race, Gender and the Body in Beauty Service Work . Berkeley, CA: University of California Press.
- Jeung, R. (2004). Faithful Generations: Race and New Asian American Churches . New Brunswick, NJ: Rutgers University Press.
- Lee, J. & Zhou, M. (Eds.). (2004). Asian American Youth: Culture, Identity, and Ethnicity . New York, NY: Routledge.
- Leong, R. (1995). Asian American Sexualities: Dimensions of the Gay and Lesbian Experience . New York, NY: Routledge.
- Ling, H. (2007). Voices of the Heart: Asian American Women on Immigration, Work, and Family . Kirksville, MO:Truman State University Press.
- Mishima, Y. (1988). Confessions of a Mask . New York, NY: W.W. Norton & Company.
- Prasso, S. (2006). The Asian Mystique: Dragon Ladies, Geisha Girls, and Our Fantasies of the Exotic Orient . New York, NY: Public Affairs Publishing.
- Quang, B., Yanagihara, H. & Liu, T. (Eds.). (2000). Take Out: Queer Writing from Asian Pacific America . New York, NY: Asian American Writers' Workshop.
- Seagrave, S. (1992). Dragon Lady: The Life and Legend of the Last Empress of China. New York, NY: Knopf Books.
- Seidman, S. (2002). Beyond the Closet: The Transformation of Gay and Lesbian Life . New York, NY: Routledge.
- Shimizu, C. (2007). The Hypersexuality of Race: Performing Asian/American Women on Screen and Scene . Durham, NC: Duke University Press.
- Stevenson, M.R. (2003). Everyday Activism: A Handbook for Lesbian, Gay, and Bisexual People and Their Allies . New York, NY: Routledge.
- Tan, J. (1998).  Jersey City, NJ: Cleis Press.
- Toyama, N.A., Gee, T., Khang, K., de Leon, C.H., & Dean, A. (Eds.). (2005). More Than Serving Tea: Asian American Women on Expectations, Relationships, Leadership and Faith . Westmont, IL: IVP Books.
- Valverde, K., Linh, C. & Wei Ming, D. (Eds.). (2019). Fight the Tower: Asian American Women Scholars’ Resistance and Renewal in the Academy . New Brunswick, NJ: Rutgers University Press.
- Wat, E.C. (2002). Making of a Gay Asian Community: An Oral History of Pre-AIDS Los Angeles . New York, NY: Rowman and Littlefield.


16.1: CORRECTIONS
CHAPTER 16 - CORRECTIONS, PRISONERS’ RIGHTS, AND PAROLE
CORRECTIONS [154]
REENTRY AND THE REVOLVING DOOR
Parole, as discussed in the previous chapter, has had mixed reviews. Overall, the effectiveness of parole hovers around 50% success. It is estimated somewhere between 600,000 and 800,000 parolees are on parole in any given year over the last 3 decades. Additionally, several hundred thousand are exiting parole in each of these years. This brings up questions about what happens to these individuals. The reality is that most of them will be rearrested. In one of the more comprehensive studies on recidivism, Alper, Durose, and Markman (2018) discussed the recidivism rate of individuals tracked over a 9-year follow up period. What they found was that rearrest occurred for about 70% in the first three years, and by year 9, 83% of the individuals released has been rearrested. Many of these individuals return to prison, hence the concept of the “revolving door” of justice. In order for reentry programs to be more successful, individuals returning to society need assistance to get back on their feet and stay on their feet. This includes items such as education and training, employment assistance to get a job, legal services, education on public benefits, and housing assistance. Interestingly, it appears as though many of the items here are the same items that many of them had deficits in that landed them in trouble in their lives before. That is – many of these items are those same predictors of offending that were discussed in the first section (known as the know predictors of recidivism). Unfortunately, it appears as though they are not getting these while they are incarcerated. Again, creating a cycle of release and catch again.
Situations and circumstances that compound these problems for many ex-offenders is the difficulty faced with trying to get a job once released. Over the last 20 years, there was an overwhelming push to include items on employment applications that asked questions about prior incarceration history. Not only were there questions about prior incarcerations and prior convictions, but many employers also have questions about ever being arrested. If an individual told the truth (which is what they should do), the reality is that their applications would be discarded, or overlooked for others without an arrest/conviction. If an ex-offender lied about it, and it was discovered during a background investigation, the application was certainly discarded. In either scenario, it became increasingly difficult for an individual to obtain legitimate employment.
This is also true on apartment rental applications. Again, when individuals would put down prior arrests, their applications would often be placed at the bottom of the pile. If someone were to lie about it, and it was discovered, it could be used as grounds for not selecting an individual for tenancy. Once again, society was making it difficult for ex-offenders to even function as a normal citizen, based on a sentence that they had served, which is when the punishment should have ended. Collectively, these items are included in the concept of collateral consequences. That is – items that are barriers to successful integration that are remnants of prior punishment.
FUTURE OUTLOOK OF CORRECTIONS
Based on the major issues presented, overcrowding and reentry, the problems faced in corrections are not likely to go away anytime soon. We have seen an increase in the overall correctional population for years now. While there are some reductions in prisons, this is not likely to stay this way, unless changes are made. Additionally, while there is space for growth in the area of community corrections, the functions of CC need to be supported and done based on evidence-based practices if it is to be more successful. It too has limits, and without the support, it is more likely to be another failure. If it is not supported, then the prison population is likely to increase even more, due to the eventual placement of too many failures of individuals in community corrections. Most offenders are in need of some basic assistance to get themselves back to a functioning level in society, including addressing their education, their substance abuse, their employment, and general and mental health. Our correctional system needs to change its habit of treating substance abuse and mental health issues as legal and punish-oriented issues if we are going to curb the tide of the growing problems we face in corrections. If not, our 8 million individuals in all forms of correctional control can quickly turn 10 million. According to a 2016 report from the U.S. Department of Education (p. 13), “from 1979–80 to 2012–13, state and local government expenditures on corrections rose by 324 percent (from $17billion to $71 billion).” Keep in mind that is taxpayer money. We are funding this issue. It is time to address these problems from a more holistic approach if we are going to see a change in our current correctional practices.
HISTORY OF CORRECTIONAL FACILITIES [155]
Prior to the 1800s, common law countries relied heavily on physical punishments. Influenced by the high ideas of the enlightenment, reformers began to move the criminal justice system away from physical punishments in favor of reforming offenders. This was a dramatic shift away from the mere infliction of pain that had prevailed for centuries. Among these early reformers was John Howard , who advocated the use of penitentiaries . Penitentiaries, as the name suggests, were places for offenders to be penitent . That is, they would engage in work and reflection on their misdeeds. To achieve the appropriate atmosphere for penitence, prisoners were kept in solitary cells with much time for reflection.
Philadelphia’s Walnut Street Jail was an early effort to model the European penitentiaries. The system used there later became known as the Pennsylvania System . Under this system, inmates were kept in solitary confinement in small, dark cells. A key element of the Pennsylvania System is that no communications whatsoever were allowed. Critics of this system began to speak out against the practice of solitary confinement early on. They maintained that the isolated conditions were emotionally damaging to inmates, causing severe distress and even mental breakdowns. Nevertheless, prisons across the United States began adopting the Pennsylvania model, espousing the value of rehabilitation.
The New York system evolved along similar lines, starting with the opening of New York’s Auburn Penitentiary in 1819. This facility used what came to be known as the congregate system . Under this system, inmates spent their nights in individual cells, but were required to congregate in workshops during the day. Work was serious business, and inmates were not allowed to talk while on the job or at meals. This emphasis on labor has been associated with the values that accompanied the Industrial Revolution. By the middle of the nineteenth century, prospects for the penitentiary movement were grim. No evidence had been mustered to suggest that penitentiaries had any real impact on rehabilitation and recidivism.
Prisons in the South and West were quite different from those in the Northeast. In the Deep South, the lease system developed. Under the lease system, businesses negotiated with the state to exchange convict labor for the care of the inmates. Prisoners were primarily used for hard, manual labor, such as logging, cotton picking, and railroad construction. Eastern ideas of penology did not catch on in the West, with the exception of California. Prior to statehood, many frontier prisoners were held in federal military prisons.
Disillusionment with the penitentiary idea, combined with overcrowding and understaffing, led to deplorable prison conditions across the country by the middle of the nineteenth century. New York’s Sing Prison was a noteworthy example of the brutality and corruption of that time. A new wave of reform achieved momentum in 1870 after a meeting of the National Prison Association (which would later become the American Correctional Association). At this meeting held in Cincinnati, members issued a Declaration of Principles. This document expressed the idea that prisons should be operated according to a philosophy that prisoners should be reformed, and that reform should be rewarded with release from confinement. This ushered in what has been called the Reformatory Movement .
One of the earliest prisons to adopt this philosophy was the Elmira Reformatory , which was opened in 1876 under the leadership of Zebulon Brockway . Brockway ran the reformatory in accordance with the idea that education was the key to inmate reform. Clear rules were articulated, and inmates that followed those rules were classified at higher levels of privilege. Under this “mark” system, prisoners earned marks (credits) toward release. The number of marks that an inmate was required to earn in order to be released was established according to the seriousness of the offense. This was a movement away from the doctrine of proportionality, and toward indeterminate sentences and community corrections.
The next major wave of corrections reform was known as the rehabilitation model , which achieved momentum during the 1930s. This era was marked by public favor with psychology and other social and behavioral sciences. Ideas of punishment gave way to ideas of treatment, and optimistic reformers began attempts to rectify social and intellectual deficiencies that were the proximate causes of criminal activity. This was essentially a medical model in which criminality was a sort of disease that could be cured. This model held sway until the 1970s when rising crime rates and a changing prison population undermined public confidence.
After the belief that “nothing works” became popular, the crime control model became the dominate paradigm of corrections in the United States. The model attacked the rehabilitative model as being “soft on crime.” “Get tough” policies became the norm throughout the 1980s and 1990s, and lengthy prison sentences became common. The aftermath of this has been a dramatic increase in prison populations and a corresponding increase in corrections expenditures. Those expenditures have reached the point that many states can no longer sustain their departments of correction. The pendulum seems to be swinging back toward a rehabilitative model, with an emphasis on community corrections. While the community model has existed parallel to the crime control model for many years, it seems to be growing in prominence.
PRISON TYPES [156]
Prisons in the United States today are usually distinguished by custody levels . Super-maximum-security prisons are used to house the most violent and most escape prone inmates. These institutions are characterized by almost no inmate mobility within the facility, and fortress-like security measures. This type of facility is very expensive to build and operate. The first such prison was the notorious federal prison Alcatraz, built by the Federal Bureau of Prisons in 1934.
Maximum-security prisons are fortresses that house the most dangerous prisoners. Only 20% of the prisons in the United States are labeled as maximum security, but, because of their size, they hold about 33% of the inmates in custody. Because super-max prisons are relatively rare, maximum-security facilities hold the vast majority of America’s dangerous convicts. These facilities are characterized by very low levels of inmate mobility, and extensive physical security measures. Tall walls and fences are common features, usually topped with razor wire. Watchtowers staffed by officers armed with rifles are common as well. Security lighting and video cameras are almost universal features.
 These areas are usually segregated from the general population, and extra security measures are put in place. Death row is often regarded as a prison within a prison, often having different staff and procedures than the rest of the facility.
Medium-security prisons use a series of fences or walls to hold prisoners that, while still considered dangerous, are less of a threat than maximum-security prisoners. The physical security measures placed in these facilities is often as tight as for maximum-security institutions. The major difference is that medium-security facilities offer more inmate mobility, which translates into more treatment and work options. These institutions are most likely to engage inmates in industrial work, such as the printing of license plates for the State.
Minimum-security prisons are institutions that usually do not have walls and armed security. Prisoners housed in minimum-security prisons are considered to be nonviolent and represent a very small escape risk. Most of these institutions have far more programs for inmates, both inside the prison and outside in the community. Part of the difference in inmate rights and privileges stems from the fact that most inmates in minimum-security facilities are “short timers.” In other words, they are scheduled for release soon. The idea is to make the often-problematic transition from prison to community go more smoothly. Inmates in these facilities may be assigned there initially, or they may have worked their way down from higher security levels through good behavior and an approaching release date.
Women are most often housed in women’s prisons . These are distinguished along the same lines as male institutions. These institutions tend to be smaller than their male counterparts are, and there are far fewer of them. Women do not tend to be as violent as men are, and this is reflected in what they are incarcerated for. The majority of female inmates are incarcerated for drug offenses. Inmate turnover tends to be higher in women’s prisons because they tend to receive shorter sentences.
A few states operate coeducational prisons where both male and female inmates live together. The reason for this is that administrators believe that a more normal social environment will better facilitate eventual reintegration of both sexes into society. The fear of predation by adult male offenders keeps most facilities segregated by gender.
In the recent past, the dramatic growth in prison populations led to the emergence of private prisons. Private organizations claimed that they could own and operate prisons more efficiently than government agencies can. The Corrections Corporation of America is the largest commercial operator of jails and prisons in the United States. The popularity of the idea has waned in recent years, mostly due to legal liability issues and a failure to realize the huge savings promised by the private corporations.
Prisons in the United States can also be parceled out by jurisdiction and by intensity. By jurisdiction, this is referring to who manages the prisons. A prison warden is generally considered the managerial face of the institution. However, a prison warden and the prison itself is usually within a much larger organizational structure. Although not always, these are usually separated by State. There are a few jurisdictions not at the State level that manage or operate prisons. This includes places like New York, Chicago, Philadelphia, and Washington D.C. Puerto Rico (not a State) also has a prison, as does the U.S. territory of the Virgin Islands.
STATE PRISONS
The normal label for the organizational structure of prisons in a particular State is often called Departments of Corrections and are run by a Director, who is usually appointed by a Governor. For example, in Oregon, it is the Oregon Department of Corrections, and Director Peters is the current head of this organization (2012-present). The Oregon Department of Corrections currently overseas 14 State prisons. More information about the ODOC can be found here: ODOC - Oregon Department of Corrections . California has the California Department of Corrections, and Secretary Diaz is the head of this organization (2018 to present). CDCR oversees 34 adult institutions. For more information about CDCR, see CDCR .
FEDERAL PRISONS
The Federal Bureau of Prisons was established in the early 1930s as a result of the need to house an increasing number of individuals convicted of federal crimes. There were already some federal prisons in place, but it was not until 1930 that the U.S. Congress passed legislation to create the BOP, housing it under the justice department. Sanford Bates became the first Director of the Federal Bureau of Prisons (FBOP or BOP), based on his long-standing work as an organizer and leader at Elmira Reformatory in New York. As more federal legislation was passed, the need for more prisons became apparent.
Today, the BOP has 109 prisons, along with numerous additional facilities (camps) adjoining at these locations. There are also military prisons, and alternative facilities, reentry centers, and training centers, that are managed by the BOP. The federal prisons are separated into six regions., which include the Mid-Atlantic Region, the North Central Region, the Northeast Region, the Southeast Region, the South Central Region, and the Western Region.
Within these regions are regional directors, which is similar to state-level directors of departments of corrections. Below is a detailed map of the regions of the Federal Bureau of Prisons. As is depicted, there are several different types of facilities within each region. A central office is also designated for each of the six regions. Click on the link in the annotation of the map to see it in a larger scale.
Table 16.1 FBOP Regional map [157]
PRIVATE PRISONS
The privatization of goods and services has long been a staple of state departments of corrections, as it allows these organizations to subcontract specific tasks within their prisons. This includes services like food and transportation services, medical, dental, and mental health services, education services, even laundry services. As mentioned in the previous section on punishment, there was much ado about crime in the United States in the 1970s and 1980s. This brought on an increased fear of crime and a more punitive state within the United States. It was during this time that a small company known as Wackenhut, a subsidiary of The Wackenhut Corporation (TWC) sought to privatize the entirety of a prison, not just services within the prison. A second company, Corrections Corporation of America ultimately won the contract and became the first privately owned prison in the United States (1984). Today, Core Civic (formerly Corrections Corporation of America) runs approximately 128 facilities in the United States. The GEO Group, the other primary private prison company runs 136 correctional, detention, or reentry facilities. Pictured below, roughly half of the 50 States in America use private prison industry prisons.
STATES USING PRIVATE PRISON INDUSTRY
Table 16.2 Private Prison State Map [158]
Much debate has come from the incorporation of private prisons. The critics of private prisons denote the lack of transparency in the reporting processes that would come from a normal prison. Still, others tackle a bigger issue – punishment for profit. That is – while taxpayers ultimately pay for all punishment of individuals, either at the State or Federal level, shareholders and administrators of the companies are making money by punishing people, under the guise of capitalism. For a more in-depth review of this, see a report presented by the Sentencing Project https://www.sentencingproject.org/publications/private-prisons-united-states/ and on NPR: https://www.npr.org/2010/10/28/130833741/prison-economics-help-drive-ariz-immigration-law
PRISONER’S RIGHTS [159]
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Learning Objective: Key Terms |
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American courts were reluctant to get involved in prison affairs during most of the 19 th century. Until the 1960s, the courts used a hands-off approach to dealing with corrections. Since then, the court has recognized that “Prison walls do not form a barrier separating prison inmates from the protections of the Constitution” (Turner v. Safley, 1987). Prisoners do give up certain rights because of conviction, but not all of them. The high courts have established that prisoners retain certain constitutional rights. As the Court stated in Hudson v. Palmer (1984), “While prisoners enjoy many protections of the Constitution that are not fundamentally inconsistent with imprisonment itself or incompatible with the objectives of incarceration, imprisonment carries with it the circumscription or loss of many rights as being necessary to accommodate the institutional needs and objectives of prison facilities, particularly internal security and safety.” From this statement, it can be seen that institutional safety and security will usually trump inmate rights when the two collide in Court.
POLITICAL RIGHTS
The phrase “ political right ” is used to refer to rights related to the participation in the democracy of the United States. Chief among these is the right to vote. The Constitution of the United States allows states to revoke a person’s right to vote upon conviction but does not require it. Several states revoke the right to vote while a person is incarcerated but restore the right once the person is released from prison. A few states revoke the right to vote for life when a person is convicted of a felony. The right to vote cannot be denied to those who are pretrial detainees confined to jail or a misdemeanant. These individuals are usually given the right to vote by absentee ballot.
THE RIGHT TO FREE SPEECH AND ASSEMBLY
The First Amendment right of prisoners to free speech is curtailed, but not eliminated. Prison administrators must justify restrictions on free speech rights. The rights to assemble is generally curtailed. As a rule, prison administrators can ban any inmate activity that is a risk to the security and safety of the institution.
THE RIGHT TO FREEDOM OF RELIGION
Generally, prisoners have the right to free exercise of their religious beliefs. These, however, can be curtailed when the health and safety of the institution are at risk. To be protected, the particular religious beliefs must be “sincerely held.” Prison officials may not, however, legally show preference for one religion over another. In practice, some religious customs have conflicted with prison policies, such as requiring work on religious holidays that forbid labor. These types of policies have been upheld by the courts.
THE RIGHT OF ACCESS TO THE COURTS
The First Amendment guarantees the right “to petition the Government for a redress of grievances.” For prisoners, this has translated to certain types of access to the courts. The two major categories of petitions that can be filed by prisoners are criminal appeals (often by habeas corpus petitions) and civil rights lawsuits. The right to petition the courts in these ways is referred to as the right of access to the courts . The court discusses this right at length in the case of Johnson v. Avery (1969).
FREEDOM FROM RETALIATION
Inmates who file complaints, grievances, and lawsuits against prison staff have a constitutional right to be free from retaliation. The Supreme Court based this right on the logic that retaliation by prison staff hampers the exercise of protected constitutional rights. In practice, this right has been difficult for inmates to assert. Prison staff can often find legitimate reasons for taking action that was intended as retaliation.
RIGHTS DURING PRISON DISCIPLINARY PROCEEDINGS
In the landmark case of Wolff v. McDonnell (1974), the Supreme Court defined the contours of prisoner rights during prison disciplinary proceedings. While not all due process rights due a criminal defendant were due the prisoner in a disciplinary proceeding, some rights were preserved. Among those rights were:
- Advance written notice of charges must be given to the disciplinary action inmate, no less than 24 hours before his appearance before the Adjustment Committee.
- There must be a written statement by the factfinders as to the evidence relied on and reasons for the disciplinary action.
- The inmate should be allowed to call witnesses and present documentary evidence in his defense if permitting him to do so will not jeopardize institutional safety or correctional goals.
- The inmate has no constitutional right to confrontation and cross-examination in prison disciplinary proceedings, such procedures in the current environment, where prison disruption remains a serious concern, being discretionary with the prison officials.
- Inmates have no right to retained or appointed counsel.
THE RIGHT TO PRIVACY
The right to privacy is closely related to the law of search and seizure. In the landmark case of Hudson v. Palmer (1984), the Court determined that inmates do not have a reasonable expectation of privacy in their living quarters. In the Court’s rationale, the needs of institutional security outweigh the inmate’s right to privacy. The policy implication of this decision is that shakedowns may be conducted at the discretion of prison staff, and no evidence of wrongdoing is necessary to justify the search.
THE RIGHT TO BE FREE FROM CRUEL AND UNUSUAL PUNISHMENT
The right to be free from cruel and unusual punishment as guaranteed by the Eighth Amendment to the United States Constitution.
Conditions in prison must not involve the “wanton and unnecessary” infliction of pain. Prison conditions, taken alone or in combination, may deprive inmates of the “minimal civilized measure of life’s necessities.” If this happens, the Court will judge the conditions of confinement unconstitutional. Conditions that cannot be said to be cruel and unusual under “contemporary standards” are not unconstitutional. According to the Court, prison conditions that are “restrictive and even harsh,” are part of the penalty that criminal offenders pay for their “offenses against society” ( Rhodes v. Chapman , 1981).
In Estelle v. Gamble (1976), the court ruled that “Deliberate indifference by prison personnel to a prisoner’s serious illness or injury constitutes cruel and unusual punishment contravening the Eighth Amendment.”
PAROLE [160]
While the process to get onto parole is unique to all of the other community sanctions we have discussed so far in this section, individuals on parole are in the community. Thus, parole is often placed within the concepts of community corrections. Parole is the release (under conditions) of an individual after they have served a portion of their sentence. It is also accompanied by the threat of re-incarceration if warranted. As with most concepts in our legal system, their roots of parole can be traced back to concepts from England and Europe. However, parole today has greatly evolved based on American values and concepts. Parole in the United States began as a concept at the first American Prison Association meeting in 1870. There was much support for the ideals of reform in corrections in America at the time. Advocates for reform helped to create the concept of parole and how it would look in the U.S., and plans to develop parole went from there. Parole authorities began establishing within the States, and by the mid-1940s, all States had a parole authority. Parole Boards and State parole authorities have fluctuated over the years, but the concept is still practiced, in varying degrees today. It is different than probation, which often operates under the judicial branch. Parole typically operates under the executive branch and is aligned with the departments of corrections, as parole is a direct extension of prison terms and release. Many states operate a post-prison supervision addendum to their sentencing matrix for the punishment of individuals.
OREGON SENTENCING GUIDELINES
Table 16.3 Oregon Sentencing Guidelines Grid [161]
As you can see from the graph, the PPS section in gray represents the recommended times for parole (post-prison supervision). Today, there are three basic types of parole in the United States, discretionary, mandatory, and expiatory.
Discretionary parole is when an individual is eligible for parole or goes before a parole board prior to their mandatory parole eligibility date. It is at the discretion of the parole board to grant parole (with conditions) for these individuals. These prisoners are generally well-behaving prisoners that have demonstrated they can function within society (have completed all required programming). Discretionary parole had seen a rapid increase in the 1980s but took a marked decrease starting in the early 1990s. In more recent years, it is continuing to return as a viable release mechanism for over 100,000 inmates a year.
Mandatory parole occurs when a prisoner hits a particular point in time in their sentence. When an inmate is sent to prison, two clocks begin. The first clock is forward counting and continues until their last day. The second clock starts at the end of their sentence and starts to work backward proportional to the “good days” an inmate has. Good days are days that an offender is free from incidents, write-ups, tickets, or other ways to describe rule infractions. For instance, for every week that an offender is a good prisoner, they might get two days taken off of the end of their sentence. When these two times converge, that would be a point in which mandatory parole could kick in for them. This must also be conditioned by truth in sentencing legislation, or what is considered an 85% rule. Many states have laws in place that stipulates that an inmate is not eligible for mandatory parole until they hit 85% of their original sentence. Thus, even though the date for the good days would be before the 85% of a sentence is served, they would only be eligible for mandatory parole once they had achieved 85% of their sentence. Recently, States have begun to soften these 85% rules, as another valve to reduce crowding issues. The Hughes et al. (2001) article also provides their proportions, indicating a direct inverse relationship to discretionary parole during the 1990s. As discretionary parole went down, mandatory parole went up. This is logical though, as once they had passed a date for discretionary parole, the next date would be an inmate’s mandatory parole date. As you can see from the image below, these proportions of releases switched in the 1990s.
PAROLE RELEASES
Table 16.4 Parole Releases [162]
Perhaps most troubling is the Expiatory Release. We see a slow increase of expiatory release in the chart, and this has continued to climb in the 2000s. Expiatory release means that a person has served their entire sentence length (and sometimes more). Based on the need to release individuals to accommodate incoming prisoners, this usually means that an inmate has misbehaved enough to nullify their “good days.” This is unfortunate, because of the three types of release, it could be argued that these are the inmates that need the most post-prison supervision. And yet, these are the inmates that are typically receiving the smallest amounts of parole.
PAROLE SUCCESS
It should again come as no surprise as to the effectiveness of parole, considering how many of the other community-based sanctions are operating. Successful parole completion rates hover around 50%, given a particular year. In the Hughes et al. (2001) article just mentioned, successful completion was roughly 42% in 1999. The same issues for failure that are found in probation completion are found in parole completion, to include: revocation failures, new charges, absconding, and other infractions. This lower-than-expected success rate has prompted many critics to argue parole. It is suggested that we are being too lenient on some while keeping lower-level inmates in prison too long. It is also argued that we are releasing dangerous individuals out into the community. Whatever the criticisms are, it is certain that we are bound to use parole as a function of release, even if it is only on paper. For example, California has a concept called non-revocable parole. The basic premise of this is: as long as you do not violate your terms of parole, your parole will be solely on paper, with no parole office check-ins. Additionally, no one will come out to your dwelling to monitor you. Effectively, this version of parole is not enforceable, hence why it is considered as parole on paper only. But the questions around parole still remain. What are we to do with the hundreds of thousands of offenders we let out of prison each year? Do they need more assistance than a bus ticket back to their county of residence? How should we be doing parole in the United States? A more modern term for parole is called re-entry. The next section covers current issues within corrections, to include what we do for inmates who are re-entering society.