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Cheat Sheet

MATH 126

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

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.