align fix; theorem style; condition number

quarto/integral_vector_calculus/review.qmd

@@ -99,12 +99,13 @@ In dimension $m=3$, the **binormal** vector, $\hat{B}$, is the unit vector $\hat
 The [Frenet-Serret]() formulas define the **curvature**, $\kappa$,  and the **torsion**, $\tau$, by
 
 
-
+$$
 \begin{align*}
 \frac{d\hat{T}}{ds} &=  & \kappa \hat{N} &\\
 \frac{d\hat{N}}{ds} &= -\kappa\hat{T} & & + \tau\hat{B}\\
 \frac{d\hat{B}}{ds} &= & -\tau\hat{N}&
 \end{align*}
+$$
 
 
 These formulas apply in dimension $m=2$ with $\hat{B}=\vec{0}$.
@@ -122,13 +123,14 @@ The chain rule says $(\vec{r}(g(t))' = \vec{r}'(g(t)) g'(t)$.
 A scalar function, $f:R^n\rightarrow R$, $n > 1$ has a **partial derivative** defined. For $n=2$, these are:
 
 
-
+$$
 \begin{align*}
 \frac{\partial{f}}{\partial{x}}(x,y) &=
 \lim_{h\rightarrow 0} \frac{f(x+h,y)-f(x,y)}{h}\\
 \frac{\partial{f}}{\partial{y}}(x,y) &=
 \lim_{h\rightarrow 0} \frac{f(x,y+h)-f(x,y)}{h}.
 \end{align*}
+$$
 
 
 The generalization to $n>2$ is clear - the partial derivative in $x_i$ is the derivative of $f$ when the *other* $x_j$ are held constant.
@@ -356,7 +358,7 @@ $$
 In two dimensions, we have the following interpretations:
 
 
-
+$$
 \begin{align*}
 \iint_R dA &= \text{area of } R\\
 \iint_R \rho dA &= \text{mass with constant density }\rho\\
@@ -364,12 +366,13 @@ In two dimensions, we have the following interpretations:
 \frac{1}{\text{area}}\iint_R x \rho(x,y)dA &= \text{centroid of region in } x \text{ direction}\\
 \frac{1}{\text{area}}\iint_R y \rho(x,y)dA &= \text{centroid of region in } y \text{ direction}
 \end{align*}
+$$
 
 
 In three dimensions, we have the following interpretations:
 
 
-
+$$
 \begin{align*}
 \iint_VdV &= \text{volume of } V\\
 \iint_V \rho dV &= \text{mass with constant density }\rho\\
@@ -378,6 +381,7 @@ In three dimensions, we have the following interpretations:
 \frac{1}{\text{volume}}\iint_V y \rho(x,y)dV &= \text{centroid of volume in } y \text{ direction}\\
 \frac{1}{\text{volume}}\iint_V z \rho(x,y)dV &= \text{centroid of volume in } z \text{ direction}
 \end{align*}
+$$
 
 
 To compute integrals over non-box-like regions, Fubini's theorem may be utilized. Alternatively, a **transformation** of variables