Fix some typos in doc for step-71

diff --git a/examples/step-71/doc/intro.dox b/examples/step-71/doc/intro.dox
index 1ad1cfa8b9f9d0081cdc02fb7cbecdb0a76a80ac..b27d7d77d501bfc6c1f86b8fa41fff1b266dc623 100644 (file)
--- a/examples/step-71/doc/intro.dox
+++ b/examples/step-71/doc/intro.dox
@@ -331,7 +331,7 @@ multi-variate function returns the derivative of that function with respect
 to one of those variables while holding the others constant:
 @f[
   \frac{\partial f\left(x, y\right)}{\partial x}
-  = \frac{d f\left(x, y\right)}{d x} \vert_{y} .
+  = \frac{d f\left(x, y\right)}{d x} \Big\vert_{y} .
 @f]
 More specific to what's encoded in the dissipation inequality (with the very general
 free energy density function $\psi_{0}$ with its parameterization yet to be formalized),
@@ -341,10 +341,10 @@ derivative would imply judicious use of the chain rule. This can be better
 understood by comparing the following two statements:
 @f{align*}
   \frac{\partial f\left(x, y\left(x\right)\right)}{\partial x}
-  &= \frac{d f\left(x, y\left(x\right)\right)}{d x} \vert_{y} \\
+  &= \frac{d f\left(x, y\left(x\right)\right)}{d x} \Big\vert_{y} \\
   \frac{d f\left(x, y\left(x\right)\right)}{d x}
-  &= \frac{d f\left(x, y\left(x\right)\right)}{d x} \vert_{y}
-   + \frac{d f\left(x, y\left(x\right)\right)}{d y} \vert_{x} \frac{d y\left(x\right)}{x} .
+  &= \frac{d f\left(x, y\left(x\right)\right)}{d x} \Big\vert_{y}
+   + \frac{d f\left(x, y\left(x\right)\right)}{d y} \Big\vert_{x} \frac{d y\left(x\right)}{x} .
 @f}
 
 Returning to the thermodynamics of the problem, we next exploit the arbitrariness
@@ -406,18 +406,18 @@ these quantities may be computed by
 For the case of rate-dependent materials, this expands to
 @f{align*}{
   \mathcal{H}^{\text{tot}} \left( \mathbf{C}, \mathbf{C}_{v}^{i}, \boldsymbol{\mathbb{H}} \right)
-  &= 2 \frac{d^{2} \psi_{0} \left( \mathbf{C}, \mathbf{C}_{v}^{i}, \boldsymbol{\mathbb{H}} \right)}{d \mathbf{C} \otimes \partial \mathbf{C}} , \\
+  &= 4 \frac{d^{2} \psi_{0} \left( \mathbf{C}, \mathbf{C}_{v}^{i}, \boldsymbol{\mathbb{H}} \right)}{\partial \mathbf{C} \otimes d \mathbf{C}} , \\
   \mathbb{D} \left( \mathbf{C}, \mathbf{C}_{v}^{i}, \boldsymbol{\mathbb{H}} \right)
-  &= -\frac{d^{2} \psi_{0} \left( \mathbf{C}, \mathbf{C}_{v}^{i}, \boldsymbol{\mathbb{H}} \right)}{d \boldsymbol{\mathbb{H}} \otimes \partial \boldsymbol{\mathbb{H}}} , \\
+  &= -\frac{d^{2} \psi_{0} \left( \mathbf{C}, \mathbf{C}_{v}^{i}, \boldsymbol{\mathbb{H}} \right)}{\partial \boldsymbol{\mathbb{H}} \otimes d \boldsymbol{\mathbb{H}}} , \\
   \mathfrak{P}^{\text{tot}} \left( \mathbf{C}, \mathbf{C}_{v}^{i}, \boldsymbol{\mathbb{H}} \right)
-  &= - 2 \frac{d^{2} \psi_{0} \left( \mathbf{C}, \mathbf{C}_{v}^{i}, \boldsymbol{\mathbb{H}} \right)}{d \boldsymbol{\mathbb{H}} \otimes \partial \mathbf{C}} , \\
+  &= - 2 \frac{d^{2} \psi_{0} \left( \mathbf{C}, \mathbf{C}_{v}^{i}, \boldsymbol{\mathbb{H}} \right)}{\partial \boldsymbol{\mathbb{H}} \otimes d \mathbf{C}} , \\
   \left[ \mathfrak{P}^{\text{tot}} \left( \mathbf{C}, \mathbf{C}_{v}^{i}, \boldsymbol{\mathbb{H}} \right)  \right]^{T}
-  &= - 2 \frac{d^{2} \psi_{0} \left( \mathbf{C}, \mathbf{C}_{v}^{i}, \boldsymbol{\mathbb{H}} \right)}{d \mathbf{C} \otimes \partial \boldsymbol{\mathbb{H}}} ,
+  &= - 2 \frac{d^{2} \psi_{0} \left( \mathbf{C}, \mathbf{C}_{v}^{i}, \boldsymbol{\mathbb{H}} \right)}{\partial \mathbf{C} \otimes d \boldsymbol{\mathbb{H}}} ,
 @f}
 while for rate-independent materials the linearizations are
 @f{align*}{
   \mathcal{H}^{\text{tot}} \left( \mathbf{C}, \boldsymbol{\mathbb{H}} \right)
-  &= 2 \frac{d^{2} \psi_{0} \left( \mathbf{C}, \boldsymbol{\mathbb{H}} \right)}{d \mathbf{C} \otimes d \mathbf{C}} , \\
+  &= 4 \frac{d^{2} \psi_{0} \left( \mathbf{C}, \boldsymbol{\mathbb{H}} \right)}{d \mathbf{C} \otimes d \mathbf{C}} , \\
   \mathbb{D} \left( \mathbf{C}, \boldsymbol{\mathbb{H}} \right)
   &= -\frac{d^{2} \psi_{0} \left( \mathbf{C}, \boldsymbol{\mathbb{H}} \right)}{d \boldsymbol{\mathbb{H}} \otimes d \boldsymbol{\mathbb{H}}} , \\
   \mathfrak{P}^{\text{tot}} \left( \mathbf{C}, \boldsymbol{\mathbb{H}} \right)

diff --git a/examples/step-71/step-71.cc b/examples/step-71/step-71.cc
index bc9e23043f6fc648328b91221a9f26a2f3ce213d..95d668d757e95ca7345b1b1bddc6db0cca267dc9 100644 (file)
--- a/examples/step-71/step-71.cc
+++ b/examples/step-71/step-71.cc
@@ -1691,7 +1691,7 @@ namespace Step71
       // @cite Pelteret2018a, equation 29). Thereafter we can compute the
       // dissipative component of the energy density function; its expression
       // is stated in @cite Pelteret2018a (equation 28), which is a
-      // straight-forward extension of an energy density function formulated by
+      // straight-forward extension of an energy density function formulated in
       // @cite Linder2011a (equation 46).
       const Differentiation::SD::Expression f_mu_v_sd =
         1.0 +
@@ -1718,11 +1718,11 @@ namespace Step71
       // This means that deriving any function $f = f(\mathbf{C}, \mathbf{Q})$
       // with respect to  $\mathbf{C}$ will return partial derivatives
       // $\frac{\partial f(\mathbf{C}, \mathbf{Q})}{\partial \mathbf{C}}
-      // \Big\vert_{\mathbf{C}_{v}}$ as opposed to the total derivative
+      // \Big\vert_{\mathbf{Q}}$ as opposed to the total derivative
       // $\frac{d f(\mathbf{C}, \mathbf{Q}(\mathbf{C}))}{d \mathbf{C}} =
       // \frac{\partial f(\mathbf{C}, \mathbf{Q}(\mathbf{C}))}{\partial
-      // \mathbf{C}} \Big\vert_{\mathbf{C}_{v}} + \frac{\partial f(\mathbf{C},
-      // \mathbf{Q}(\mathbf{C}))}{\partial \mathbf{C}_{v}}
+      // \mathbf{C}} \Big\vert_{\mathbf{Q}} + \frac{\partial f(\mathbf{C},
+      // \mathbf{Q}(\mathbf{C}))}{\partial \mathbf{Q}}
       // \Big\vert_{\mathbf{C}} : \frac{d \mathbf{Q}(\mathbf{C}))}{d
       // \mathbf{C}}$.
       //