Cheat Sheet

MATH 126

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Table of contents

1. Linear Algebra
   1. Fundamentals, Operations, Products
   2. Geometric Entities
2. Vector Calculus
3. Partial Derivatives

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Linear Algebra

Fundamentals, Operations, Products

The dot product is defined as follows.

$\langle a_1, a_2, ..., a_n \rangle \cdot \langle b_1, b_2, ..., b_n \rangle = a_1 b_1 + a_2 b_2 + ... + a_n b_n$

If the dot product between two vectors $\vec{a}$ and $\vec{b}$ is 0, then the two vectors are orthogonal to each other.

$\vec{a} \cdot \vec{b} = \| \vec{a} \| \| \vec{b} \| \cos \theta \implies \cos \theta = \frac{\vec{a} \cdot \vec{b}}{\| \vec{a} \| \| \vec{b} \|}$

A vector is defined $\vec{a} = \langle a_1, a_2, a_3 \rangle$. $\alpha, \beta, \gamma$ are the angles $\vec{a}$ makes with the $x, y, z$ axes. The following are true:

$\cos \alpha = \frac{a_1}{\| \vec{a} \|}$ $\cos \beta = \frac{a_2}{\| \vec{a} \|}$ $\cos \gamma = \frac{a_3}{\| \vec{a} \|}$

The projection of a vector $\vec{a}$ onto $\vec{b}$ is $\text{proj}_{\vec{b}} \vec{a}$, defined as follows.

$\text{proj}_{\vec{b}} \vec{a} = \text{comp}_{\vec{b}} \vec{a} \left( \frac{\vec{b}}{\| \vec{b} \| } \right)$ $\text{comp}_{\vec{b}} \vec{a} = \frac{\vec{a} \cdot \vec{b}}{\| \vec{b} \|}$

The cross product between two three-dimensional vectors is defined as follows.

$\langle a_1, a_2, a_3 \rangle \times \langle b_1, b_2, b_3 \rangle = \langle a_2 b_3 - a_3 b_2, a_3b_1 - a_1 b_3, a_1b_2 - a_2b_1 \rangle$

The cross product of two vectors $\vec{a}$ and $\vec{b}$ is orthogonal to both $\vec{a}$ and $\vec{b}$.

The magnitude of the cross product is the area of a parallelogram formed by the two vectors $\vec{a}$ and $\vec{b}$.

If the cross product of two vectors is zero, they are parallel.

Geometric Entities

A line can be defined using a vector equation: $\vec{r} = \vec{r}_0 + t \vec{v}$. It can be rewritten in parametric equations and symmetric equations.

A plane is defined by $a(x - x_0) + b(y - y_0) + c(z - z_0)$ for some point $(x_0, y_0, z_0)$ and a normal vector $\langle a, b, c \rangle$.

Clinders are formed by two-dimensional cross-sections extended across a free variable.

Quadric surfaces are formed by second-degree equations across three variables. To identify a quadric surface, take traces in each dimension.

Vector Calculus

To take the derivative of a vector, simply take the element-wise derivative. All differentiation rules for single-variable differentiation apply.

Arc length is given by $L = \int_a^b \| r'(t) \| dt$, where $r(t)$ is the position vector, $r'(t)$ is the velocity vector, and $\| r'(t) \|$ is the speed. That is, distance can be computed as the integral of speed.

You can parametrize a curve with respect to arc length. Derive an equation for arc length $s$ equal to some function of $t$, then solve for $t$ in terms of $s$ and substitute in the original position vector equation.

Curvature is defined by $\kappa = \frac{\| \vec{r}'(t) \times \vec{r}''(t) \| }{\| r'(t) \|^3}$.

Tangent, Normal, and Binormal vectors:

$\vec{T}(t) = \frac{\vec{r}'(t)}{\| \vec{r}'(t) \|}$ $\vec{N}(t) = \frac{\vec{T}'(t)}{\|\vec{T}'(t)\|}$ $\vec{B}(t) = \vec{T}(t) \times \vec{N}(t)$

• The normal plane is spanned by the normal and binormal vectors. The normal vector of the normal plane is the unit tangent vector $\vec{T}$.
• The osculating plane is spanned by the unit tangent vector and the normal vectors. The normal vector of the osculating plane is the binormal vector.

Motion in Space

$\vec{v}(t) = \vec{v}(t_0) + \int_{t_0}^t \vec{a}(u) du$ $\vec{r}(t) = \vec{r}(t_0) + \int_{t_0}^t \vec{v}(u) du$ $\vec{F}(t) = m \vec{a}(t)$ $\vec{a} = a_T \vec{T} + a_n \vec{N}$ $\vec{a_T} = \nu' = \frac{\vec{r}'(t) \cdot \vec{r}''(t)}{\| \vec{r}'(t) \| }$ $\vec{a_N} = \kappa \nu^2 = \frac{\| \vec{r}'(t) \times \vec{r}''(t)\|}{\| \vec{r}'(t) \|}$

Partial Derivatives

To take the partial derivative of a function with respect to some variable $k$, treat all other variables as constants and differentiate w.r.t. $k$.

Clairaut’s Theorem. $f_{xy} (a, b) = f_{yx}(a, b)$ for two continuous functions $f_{xy}$ and $f_{yz}$.

Laplace’s equation: $\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0$. SOlutions to Laplace’s equation are harmonic functions.

The tangent plane to the surface $S$ at a point $P$ is the plane containing both tangent lines.

$z - z_0 = f_x(x_0, y_0) (x - x_0) + f_y(x_0, y_0) (y - y_0)$

To find critical values, find points at which all partial derivatives are 0 or one partial derivative does not exist. To determine the nature of the critical value, find $D$ as defined below and follow the algorithm.

\[D(a, b) = f_{xx}(a, b) f_{yy}(a, b) - \[f_{xy} (a, b)\]^2\]

1. If $D > 0$, then $f(a, b)$ is a local minimum or maximum.
2. If $f_{xx}(a, b) > 0$, then local minimum.
3. If $f_{xx}(a, b) < 0$, then local maximum.
4. If $D < 0$, then not a local maximum or minimum (saddle point).
5. If $D = 0$, then inconclusive.

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