Page Description Equation
95 function (mapping) \[ \vect{f}:X^m\rightarrow X^n \\ \textrm{where}\\ \vect{f} = \left\{ \left( \vect{x},\vect{f} \left( \vect{x} \right) \right) \in X^{m+n}:\vect{x}\in X^m \right\} \]
97 graph of a function \[ \mathrm{graph}\,f = \left\{ \left( \vect{x},f \left( \vect{x} \right) \right) \in \mathbb{R}^{n+1}:\vect{x}\in \mathbb{R}^n \right\}\\ \textrm{where}\\ f:\mathbb{R}^n \rightarrow \mathbb{R} \]
99 level set of value $c$ \[ L_c = \left\{ x\in U:f \left( \vect{x} \right)=c \right\}\\ \textrm{where}\\ f:U\subset\mathbb{R}^n\rightarrow\mathbb{R} \;\textrm{and}\; c\in\mathbb{R} \]
100 section of the graph of $f,$ the intersection of the graph and a vertical plane. \[ S_{x_i=0} = \left\{ \left( \vect{x}, f\left(\vect{x}\right) \right) \in \mathbb{R}^{n+1}:x_i=0, \vect{x} \in \mathbb{R}^n \right\}\\ \textrm{where} \ \vect{x}=\left(x_1,\ldots,x_n\right) \]
134 differentiable \[ \lim_{x\rightarrow x_0} \frac{ \left\| f(\vect{x})-f(\vect{x}_0) -\vect{T}(\vect{x}-\vect{x}_0) \right\|} {\left\|\vect{x}-\vect{x}_0\right\|} = 0\\ \ \\ \mathbf{D}f\left(\vect{x}_0\right)\equiv\vect{T} \]
135 derivative, matrix of partial derivatives of $f$ at $x_0.$ Also called the differential of $f.$ \[ \mathbf{D}f\left(\mathbf{x}_0\right) = \begin{bmatrix} \frac{\partial f_1}{\partial x_1} &\cdots &\frac{\partial f_1}{\partial x_n}\\ \vdots&&\vdots\\ \frac{\partial f_m}{\partial x_1} &\cdots &\frac{\partial f_m}{\partial x_n} \end{bmatrix} \]
128 partial derivative
132 equation of the line tangent to the graph of a function $f:\mathbb{R}\rightarrow\mathbb{R}$ at the point $x = x_0.$ Also called the linear approximation to the function $f$ at $x = x_0.$ \begin{align*} y &= L\left(x\right)\\ &= f\left(x_0\right)+f'\left(x_0\right)(x-x_0)\\ &= f\left(x_0\right) +\left[\frac{d}{dx}f\left(x_0\right)\right]\left(x-x_0\right) \end{align*}
133 equation of the plane tangent to the graph of a function $f:R^2\rightarrow\mathbb{R}$ at the point $\left(x_0,y_0\right).$ Also called the linear approximation to the function $f$ at $\left(x_0,y_0\right).$ \[ \begin{align*} z &= L\left(x,y\right)\\ &= f\left(x_0,y_0\right) + \left[ \frac{\partial f}{\partial x} \left(x_0,y_0\right) \right] \left(x-x_0\right)\\ & \qquad + \left[ \frac{\partial f}{\partial y} \left(x_0,y_0\right) \right] \left(y-y_0\right) \end{align*} \]
gradient \[ \begin{align*} \nabla f &= \mathbf{D}f\left(\vect{x}_0\right)\\ &= \left[ \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z} \right]\\ &= \left( \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z} \right)\\ &= \vect{i}\frac{\partial f}{\partial x} +\vect{j}\frac{\partial f}{\partial y} +\vect{k}\frac{\partial f}{\partial z}\\ \end{align*} \]
142 equation of a space curve (general form) \[ \begin{align*} \vect{c}\left(t\right) &= \left( x\left(t\right), y\left(t\right), z\left(t\right) \right)\\ &= x\left(t\right)\vect{i} + y\left(t\right)\vect{j} + z\left(t\right)\vect{k}\\ &= f\left(t\right)\vect{i} + g\left(t\right)\vect{j} + h\left(t\right)\vect{k} \end{align*} \] \[ \textrm{where}\\ \begin{align*} x&=f\left(t\right)\\ y&=g\left(t\right)\\ y&=h\left(t\right) \end{align*} \]
tangent vector to curve $\vect{c}\left(t\right) = x\left(t\right)\vect{i} + y\left(t\right)\vect{j} + z\left(t\right)\vect{k}$ \[ \vect{c}'\left(t\right) = \left( x'\left(t\right), y'\left(t\right), z'\left(t\right) \right) \]
148 equation of the line tangent to the curve given by $\vect{c}\left(t\right) =x\left(t\right)\vect{i} +y\left(t\right)\vect{j} +z\left(t\right)\vect{k}$ at $\vect{c}\left(t_0\right)=P_0\left(x_0,y_0,z_0\right)$ where $\vect{c}^\prime\left(t_0\right)\neq\vect{0}.$ \[ \begin{align*} \vect{l}\left(t\right) &= \vect{c}\left(t_0\right) +\vect{c}^\prime\left(t_0\right)\left(t-t_0\right)\\ x &= x_0+x^\prime\left(t_0\right)\left(t-t_0\right)\\ y &= y_0+y^\prime\left(t_0\right)\left(t-t_0\right)\\ z &= z_0+z^\prime\left(t_0\right)\left(t-t_0\right) \end{align*} \]
velocity and position are orthogonal if distance from the origin is constant. (e.g. circles, spheres, etc.) \[ \exists c\forall t: \left|\vect{r}\right| = c\Rightarrow\vect{r}\cdot\vect{r}' = 0 \]
general equation of a helix \[ \vect{r}\left(t\right) = a\cos{\left(ct\right)}\vect{i} + a\sin{\left(ct\right)}\vect{j} + b\vect{k} \]
sum rule
product rule
quotient rule
constant multiple rule
chain rule, one-variable case
chain rule
164 directional derivative of $f$ in the direction of the unit vector $\vect{u}.$ \[ \frac{d}{dt}f\left(\vect{x}+t\vect{u}\right) =\nabla f\left(\vect{x}\right)\cdot\vect{u} \]
164 $\vect{u}$ is the direction of zero change in values of $f.$ \[ \nabla f\cdot\vect{u}=0 \]
168 equation of line tangent to level curve $C=\left\{\left(x,y\right):f\left(x,y\right)=k\right\}$ at $\left(x_0,y_0\right)$ if $\nabla f\left(x_0,y_0\right)\neq\vect{0}$ \[ \nabla f\left(x_0,y_0\right)\cdot\left(x-x_0,y-y_0\right)=0 \]
167 equation of plane tangent to level surface $S=\left\{\left(x,y,z\right):f\left(x,y,z\right)=k\right\}$ at $\left(x_0,y_0,z_0\right)$ if $\nabla f\left(x_0,y_0,z_0\right)\neq\vect{0}$ \[ \nabla f\left(x_0,y_0,z_0\right)\cdot\left(x-x_0,y-y_0,z-z_0\right)=0 \]