1$
\begin{array}{c}
f \\
\downarrow \quad \downarrow \\
x \quad y
\end{array}
$
$\frac{\partial f}{\partial x} = f_x \quad : \quad \frac{\partial}{\partial x} \left( f(x, y) \right) \quad y = \text{constant}$
$\frac{\partial f}{\partial y} = f_y \quad : \quad \frac{\partial}{\partial y} \left( f(x, y) \right) \quad x = \text{constant}$
$\frac{\partial^2 f}{\partial x^2} = f_{xx} \quad : \quad \frac{\partial}{\partial x} \left( \frac{\partial f}{\partial x} \right) \quad y = \text{constant}$
$\frac{\partial^2 f}{\partial y^2} = f_{yy} \quad : \quad \frac{\partial}{\partial y} \left( \frac{\partial f}{\partial y} \right) \quad x = \text{constant}$
$\frac{\partial^2 f}{\partial y \partial x} = f_{yx} \quad : \quad \frac{\partial}{\partial y} \left( \frac{\partial f}{\partial x} \right) \quad x = \text{constant}$
$\frac{\partial^2 f}{\partial x \partial y} = f_{xy} \quad : \quad \frac{\partial}{\partial x} \left( \frac{\partial f}{\partial y} \right) \quad y = \text{constant}$
2$
\begin{array}{c}
f \\
| \quad | \quad | \\
x \quad y \quad z
\end{array}
$
$\frac{\partial f}{\partial x} = f_x \quad : \quad \frac{\partial}{\partial x} \left( f(x, y, z) \right) \quad y, z = \text{constant}$
$\frac{\partial f}{\partial y} = f_y \quad : \quad \frac{\partial}{\partial y} \left( f(x, y, z) \right) \quad x, z = \text{constant}$
$\frac{\partial f}{\partial z} = f_z \quad : \quad \frac{\partial}{\partial z} \left( f(x, y, z) \right) \quad x, y = \text{constant}$
$\frac{\partial^2 f}{\partial x^2} = f_{xx} \quad : \quad \frac{\partial}{\partial x} \left( \frac{\partial f}{\partial x} \right) \quad y, z = \text{constant}$
$\frac{\partial^2 f}{\partial y^2} = f_{yy} \quad : \quad \frac{\partial}{\partial y} \left( \frac{\partial f}{\partial y} \right) \quad x, z = \text{constant}$
$\frac{\partial^2 f}{\partial z^2} = f_{zz} \quad : \quad \frac{\partial}{\partial z} \left( \frac{\partial f}{\partial z} \right) \quad x, y = \text{constant}$
$\frac{\partial^2 f}{\partial y \partial x} = f_{yx} \quad : \quad \frac{\partial}{\partial y} \left( \frac{\partial f}{\partial x} \right) \quad x, z = \text{constant}$
$\frac{\partial^2 f}{\partial x \partial y} = f_{xy} \quad : \quad \frac{\partial}{\partial x} \left( \frac{\partial f}{\partial y} \right) \quad y, z = \text{constant}$
$\frac{\partial^2 f}{\partial z \partial x} = f_{zx} \quad : \quad \frac{\partial}{\partial z} \left( \frac{\partial f}{\partial x} \right) \quad x, y = \text{constant}$
$\frac{\partial^2 f}{\partial x \partial z} = f_{xz} \quad : \quad \frac{\partial}{\partial x} \left( \frac{\partial f}{\partial z} \right) \quad y, z = \text{constant}$
$\frac{\partial^2 f}{\partial z \partial y} = f_{zy} \quad : \quad \frac{\partial}{\partial z} \left( \frac{\partial f}{\partial y} \right) \quad x, y = \text{constant}$
$\frac{\partial^2 f}{\partial y \partial z} = f_{yz} \quad : \quad \frac{\partial}{\partial y} \left( \frac{\partial f}{\partial z} \right) \quad x, z = \text{constant}$
3When is $f_{xy} = f_{yx}$?
When $f_x\, and\, f_y\, are \,continously\, differentiable$
4When is $f_{xyz} = f_{zyx}$? etc
answer
5$
\begin{array}{c}
f \\
\mid \quad \mid \quad \mid \\
\begin{array}{ccc}
x & y & z \\
\mid & \mid & \mid \\
t & t & t
\end{array}
\end{array}
$
$\frac{d f}{d t} = \frac{\partial f}{\partial x} \cdot \frac{d x}{d t} + \frac{\partial f}{\partial y} \cdot \frac{d y}{d t} + \frac{\partial f}{\partial z} \cdot \frac{d z}{d t}$
6$
\begin{array}{c}
f \\
\mid \quad \mid \\
\begin{array}{cc}
x & y \\
\mid \quad \mid & \mid \quad \mid \\
s \quad t & s \quad t
\end{array}
\end{array}
$
$\frac{\partial f}{\partial s} = \frac{\partial f}{\partial x} \cdot \frac{\partial x}{\partial s} + \frac{\partial f}{\partial y} \cdot \frac{\partial y}{\partial s}$
$\frac{\partial f}{\partial t} = \frac{\partial f}{\partial x} \cdot \frac{\partial x}{\partial t} + \frac{\partial f}{\partial y} \cdot \frac{\partial y}{\partial t}$
7$
\begin{array}{c}
f \\
\mid \quad \mid \quad \mid \\
\begin{array}{ccc}
x & y & z \\
\mid \quad \mid & \mid \quad \mid & \mid \quad \mid \\
s \quad t & s \quad t & s \quad t
\end{array}
\end{array}
$
$\frac{\partial f}{\partial s} = \frac{\partial f}{\partial x} \cdot \frac{\partial x}{\partial s} + \frac{\partial f}{\partial y} \cdot \frac{\partial y}{\partial s} + \frac{\partial f}{\partial z} \cdot \frac{\partial z}{\partial s}$
$\frac{\partial f}{\partial t} = \frac{\partial f}{\partial x} \cdot \frac{\partial x}{\partial t} + \frac{\partial f}{\partial y} \cdot \frac{\partial y}{\partial t} + \frac{\partial f}{\partial z} \cdot \frac{\partial z}{\partial t}$
8$
\begin{array}{c}
f \\
\mid \quad \mid \\
\begin{array}{cc}
x & y \\
& \mid \\
& x
\end{array}
\end{array}
\quad \text{and} \quad f(x, y) \text{ is constant}
$
$\frac{d y}{d x} = -\frac{f_x}{f_y}$
9$
\begin{array}{c}
f \\
\mid \quad \mid \quad \mid \\
x \quad y \quad z \\
\phantom{x} \quad \phantom{y} \quad \mid \quad \mid \\
\phantom{x} \quad \phantom{y} \quad x \quad y
\end{array} \text{and} \quad f(x, y, z) \text{ is constant}
$
$\frac{\partial z}{\partial x} = -\frac{f_x}{f_z}$
$\frac{\partial z}{\partial y} = -\frac{f_y}{f_z}$
10What type of surface plot is ax + by + cz = d
Plane
11What type of surface plot is $(x- a)^2 + (y - b)^2 + (z - c)^2 = r^2$?
represents a sphere with center (a, b, c) and radius r
12What is the method for sketching a surface plot z = f(x, y)?
$1. \quad \text{Draw } x = 0 \quad \text{(yz-plane)}$
$\quad \text{Draw } y = 0 \quad \text{(xz-plane)}$
$\quad \text{Draw } z = 0 \quad \text{(xy-plane)}$
$2. \quad \text{Combine all three into the xyz-plane.}$
13What is the method for sketching a contour plot z = f(x, y)?
$1. \quad \text{Set } z = c.$
$2. \quad \text{Choose values of } c$
$3. \quad \text{Plot in the } xy \text{-plane.}$
$\text{Note: This can also be done in the xz or yz plane by setting } y = c \text{ or } x = c$
14What is the condition for f(x, y) to be continuous at (0, 0)?
$\lim_{(x, y) \to (0, 0)} f(x, y) = C = f(0, 0)$
15What is the method to disprove the existance of a limit?
$1. \quad \text{Check path } x = 0$
$\lim_{(x, y) \to (0, 0)} f(x, y) \text{ becomes } \lim_{y \to 0} f(y). \quad \text{Evaluate this.}$
$2. \quad \text{Check path } y = 0$
$\lim_{(x, y) \to (0, 0)} f(x, y) \text{ becomes } \lim_{x \to 0} f(x). \quad \text{Evaluate this.}$
$\text{Continue to check different paths (e.g., } x = y, \, x = y^2, \, y = x^2, \ldots) \text{ until the limit along any two paths is different.}$
$\text{This proves the limit does not exist.}$
16What is the method to prove the existance of a limit?
$1. \quad \lim_{(x, y) \to (0, 0)} f(x, y) = \lim_{R \to 0} f(R, \phi), \quad \text{where } x = R \cos \phi \text{ and } y = R \sin \phi.$
$2. \quad \text{Simplify } f(R, \phi).$
$3. \quad \text{Determine if the limit exists or does not exist.}$