Multivariable Calculus and Vector Analysis Review

Vocabulary flashcards defining fundamental vector operations, spatial geometry equations, multivariable limits, partial differentiation, directional derivatives, and extrema concepts covered in the problem set.

Vector from Point $A$ to Point $B$ ($\vec{AB}$)

A directed line segment determined by subtracting the coordinates of initial point $A$ from terminal point $B$, defined component-wise as $\langle x_B - x_A, y_B - y_A, z_B - z_A \rangle$.

Vector Magnitude ($\|\mathbf{v}\|$)

The Euclidean length of a vector $\mathbf{v} = \langle v_1, v_2, v_3 \rangle$, calculated as the square root of the sum of the squares of its components: $\sqrt{v_1^2 + v_2^2 + v_3^2}$.

Unit Vector

A vector with a magnitude of exactly $1$, obtained in the direction of a non-zero vector $\mathbf{v}$ by scalar multiplying $\mathbf{v}$ by the reciprocal of its length: $\frac{\mathbf{v}}{\|\mathbf{v}\|}$.

Vector Subtraction ($\mathbf{u} - \mathbf{v}$)

An algebraic operation between two vectors of the same dimension, performed by subtracting corresponding components: $\langle u_1 - v_1, u_2 - v_2, u_3 - v_3 \rangle$.

Dot Product ($\mathbf{u} \cdot \mathbf{v}$)

A scalar operation between two vectors computed as the sum of the products of their corresponding components: $\mathbf{u} \cdot \mathbf{v} = u_1 v_1 + u_2 v_2 + u_3 v_3$.

Angle Between Two Vectors

The angle $\theta$ (where $0 \le \theta \le \pi$) between non-zero vectors $\mathbf{u}$ and $\mathbf{v}$, determined by the formula $\cos(\theta) = \frac{\mathbf{u} \cdot \mathbf{v}}{\|\mathbf{u}\| \|\mathbf{v}\|}$.

Vector Projection ($\text{proj}_{\mathbf{v}}\mathbf{u}$)

The orthogonal projection of vector $\mathbf{u}$ onto a non-zero vector $\mathbf{v}$, computed as the scalar projection multiplied by the unit vector in the direction of $\mathbf{v}$: $\left(\frac{\mathbf{u} \cdot \mathbf{v}}{\|\mathbf{v}\|^2}\right) \mathbf{v}$.

Cross Product ($\mathbf{u} \times \mathbf{v}$)

A binary vector operation in three-dimensional space that produces a vector perpendicular to both $\mathbf{u}$ and $\mathbf{v}$, having magnitude equal to the area of the parallelogram formed by the two vectors.

Area of a Triangle Formed by Two Vectors

The geometric measure of a triangle having adjacent sides defined by vectors $\mathbf{u}$ and $\mathbf{v}$, computed as half the magnitude of their cross product: $\frac{1}{2} \|\mathbf{u} \times \mathbf{v}\|$.

Symmetric Equations of a Line

Equations describing a line in three dimensions passing through point $(x_0, y_0, z_0)$ with direction vector $\langle a, b, c \rangle$, expressed as $\frac{x - x_0}{a} = \frac{y - y_0}{b} = \frac{z - z_0}{c}$.

Normal Vector of a Plane

A non-zero vector orthogonal to every vector lying entirely in the plane, whose components $\langle A, B, C \rangle$ match the linear coefficients in the scalar equation of the plane $Ax + By + Cz = D$.

Two-Path Test for Non-Existence of Limits

A method to prove that a multivariable limit $\lim_{(x, y) \to (x_0, y_0)} f(x, y)$ does not exist by demonstrating that the function approaches different limiting values along two distinct paths to the point.

Partial Derivative ($f_x$)

The rate of change of a multivariable function $f(x, y)$ with respect to the variable $x$, calculated by applying ordinary differentiation rules while treating $y$ as a constant.

Multivariable Chain Rule

A differentiation theorem for composite functions; for $f(x, y)$ with $x = x(u, v)$ and $y = y(u, v)$, it defines partial derivatives such as $\frac{\partial f}{\partial u} = \frac{\partial f}{\partial x} \frac{\partial x}{\partial u} + \frac{\partial f}{\partial y} \frac{\partial y}{\partial u}$.

Gradient Vector ($\nabla f$)

A vector-valued operator consisting of all first-order partial derivatives of a scalar function, represented for $f(x, y, z)$ as $\langle f_x, f_y, f_z \rangle$, pointing in the direction of greatest increase.

Directional Derivative ($D_{\mathbf{u}} f$)

The instantaneous rate of change of a differentiable function $f$ at a point in the direction of a specified unit vector $\mathbf{u}$, computed as the dot product $\nabla f \cdot \mathbf{u}$.

Direction of Maximum Decrease

The direction in which a multivariable function decreases most rapidly from a given point, defined by the negative gradient vector $-\nabla f$.

Maximum Rate of Decrease

The magnitude of the fastest rate at which a multivariable function diminishes at a point, evaluated as $-\|\nabla f\|$ (or a rate of magnitude $\begin{Vmatrix}\nabla f\end{Vmatrix}$ in the direction of $-\nabla f$).

Critical Point

A point $(x_0, y_0)$ in the interior of the domain of a multivariable function $f$ where either the gradient is the zero vector ($\nabla f(x_0, y_0) = \mathbf{0}$) or at least one partial derivative fails to exist.

Saddle Point

A critical point of a smooth multivariable function that is not a local extremum, where the surface curves upward in one direction and downward in another direction.