Multivariable Calculus - Vector Calculus & Optimization Flashcards

Question 1

Partial Derivative

Accepted Answer

The derivative of a multivariable function $f(x, y, \dots)$ with respect to one variable while holding all other variables constant. Represented as $\frac{\partial f}{\partial x}$ or $f_x$.

Question 2

Gradient Vector ($\
abla f$)

Accepted Answer

A vector composed of all first-order partial derivatives of a scalar function $f(x, y, z)$:
$$\
abla f = \left\langle \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z} \ight\angle$$
It points in the direction of steepest ascent, and its magnitude $\|\
abla f\|$ represents the maximum rate of change.

Question 3

Directional Derivative

Accepted Answer

The rate of change of a scalar function $f$ in the direction of a unit vector $\mathbf{u}$. Calculated using the gradient:
$$D_{\mathbf{u}} f = \
abla f \cdot \mathbf{u}$$

Question 4

Critical Points (Multivariable)

Accepted Answer

Points $(a, b)$ in the domain of $f(x, y)$ where the gradient vector is zero ($\
abla f(a, b) = \mathbf{0}$) or where at least one partial derivative does not exist.

Question 5

Second Derivative Test (Two Variables)

Accepted Answer

For a critical point $(a, b)$, compute the discriminant $D = f_{xx}f_{yy} - (f_{xy})^2$: - $D > 0$ and $f_{xx} > 0 \implies$ Local Minimum - $D > 0$ and $f_{xx} < 0 \implies$ Local Maximum - $D < 0 \implies$ Saddle Point - $D = 0 \implies$ Inconclusive test

Question 6

Hessian Matrix

Accepted Answer

A square matrix of second-order partial derivatives of a scalar function:
$$H(f) = \begin{bmatrix} \frac{\partial^2 f}{\partial x^2} & \frac{\partial^2 f}{\partial x \partial y} \\ \frac{\partial^2 f}{\partial y \partial x} & \frac{\partial^2 f}{\partial y^2} \end{bmatrix}$$
Used to analyze local curvature and confirm extrema in $n$-dimensions.

Question 7

Lagrange Multipliers

Accepted Answer

A strategy for finding local extrema of $f(\mathbf{x})$ subject to an equality constraint $g(\mathbf{x}) = k$. Solved using the vector equation:
$$\
abla f = \lambda \
abla g \quad \	ext{and} \quad g(\mathbf{x}) = k$$
where $\lambda$ is the Lagrange multiplier.

Question 8

Double Integrals & Fubini's Theorem

Accepted Answer

A double integral calculates the volume under a surface $z = f(x, y)$ over a 2D region $R$. Fubini's Theorem states that for continuous functions, the integral can be calculated via iterated integrals in any order:
$$\iint_R f(x, y) \, dA = \int_a^b \int_c^d f(x, y) \, dy \, dx = \int_c^d \int_a^b f(x, y) \, dx \, dy$$

Question 9

Polar Coordinates Transformation

Accepted Answer

Used to simplify integration over circular regions:
$$x = r \cos\	heta, \quad y = r \sin\	heta, \quad dA = r \, dr \, d\	heta$$
Always remember the extra factor of $r$ in the area element $dA$.

Question 10

Cylindrical Coordinates Transformation

Accepted Answer

Extends 2D polar coordinates to 3D by adding the standard $z$-axis:
$$x = r \cos\	heta, \quad y = r \sin\	heta, \quad z = z$$
Volume element: $dV = r \, dz \, dr \, d\	heta$.

Question 11

Spherical Coordinates Transformation

Accepted Answer

3D coordinate system using radius $\ho$, zenith angle $\phi$, and azimuthal angle $\	heta$:
$$x = \ho \sin\phi \cos\	heta, \quad y = \ho \sin\phi \sin\	heta, \quad z = \ho \cos\phi$$
Volume element: $dV = \ho^2 \sin\phi \, d\ho \, d\phi \, d\	heta$.

Question 12

Jacobian Determinant (Change of Variables)

Accepted Answer

Scaling factor when transforming coordinates $(x, y)$ to $(u, v)$:
$$J(u, v) = \frac{\partial(x, y)}{\partial(u, v)} = \det \begin{bmatrix} \frac{\partial x}{\partial u} & \frac{\partial x}{\partial v} \\ \frac{\partial y}{\partial u} & \frac{\partial y}{\partial v} \end{bmatrix}$$
Differential area converts as $dA = |J(u, v)| \, du \, dv$.

Question 13

Vector Field

Accepted Answer

A function that maps every point $(x, y, z)$ in space to a vector $\mathbf{F}(x, y, z) = \langle P, Q, R \angle$. Examples include velocity fields, gravitational fields, and electric fields.

Question 14

Line Integral of a Vector Field

Accepted Answer

Calculates the total work done by a vector field $\mathbf{F}$ along a smooth curve $C$ parameterized by $\mathbf{r}(t)$ for $a \le t \le b$:
$$\int_C \mathbf{F} \cdot d\mathbf{r} = \int_a^b \mathbf{F}(\mathbf{r}(t)) \cdot \mathbf{r}'(t) \, dt$$

Question 15

Fundamental Theorem for Line Integrals

Accepted Answer

States that line integrals of conservative vector fields $\mathbf{F} = \
abla f$ depend only on endpoints:
$$\int_C \
abla f \cdot d\mathbf{r} = f(\mathbf{r}(b)) - f(\mathbf{r}(a))$$
Such line integrals are path-independent.

Question 16

Conservative Vector Field Test

Accepted Answer

A vector field $\mathbf{F}$ defined on a simply connected domain is conservative (path-independent) if and only if its curl is zero:
$$\	ext{curl } \mathbf{F} = \
abla \	imes \mathbf{F} = \mathbf{0}$$
For a 2D field $\mathbf{F} = \langle P, Q \angle$, this simplifies to $\frac{\partial P}{\partial y} = \frac{\partial Q}{\partial x}$.

Question 17

Divergence (\	ext{div } \mathbf{F})

Accepted Answer

A scalar measure of a vector field's rate of expansion or contraction at a point:
$$\	ext{div } \mathbf{F} = \
abla \cdot \mathbf{F} = \frac{\partial P}{\partial x} + \frac{\partial Q}{\partial y} + \frac{\partial R}{\partial z}$$
Positive divergence represents a source; negative divergence represents a sink.

Question 18

Curl (\	ext{curl } \mathbf{F})

Accepted Answer

A vector measure of the local rotation/circulation of a 3D vector field:
$$\	ext{curl } \mathbf{F} = \
abla \	imes \mathbf{F} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\ P & Q & R \end{vmatrix}$$

Question 19

Green's Theorem

Accepted Answer

Relates a closed line integral around a positively oriented curve $C$ to a double integral over the enclosed plane region $D$:
$$\oint_C (P \, dx + Q \, dy) = \iint_D \left( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \ight) dA$$

Question 20

Surface Integral of a Vector Field (Flux)

Accepted Answer

Measures the net flow of a vector field $\mathbf{F}$ across an oriented surface $S$ parameterized by $\mathbf{r}(u,v)$:
$$\iint_S \mathbf{F} \cdot d\mathbf{S} = \iint_D \mathbf{F}(\mathbf{r}(u,v)) \cdot \left( \frac{\partial \mathbf{r}}{\partial u} \	imes \frac{\partial \mathbf{r}}{\partial v} \ight) du \, dv$$

Question 21

Stokes' Theorem

Accepted Answer

Relates the surface integral of the curl of a vector field over an open surface $S$ to the line integral around its boundary curve $\partial S$:
$$\iint_S (\
abla \	imes \mathbf{F}) \cdot d\mathbf{S} = \oint_{\partial S} \mathbf{F} \cdot d\mathbf{r}$$

Question 22

Divergence Theorem (Gauss's Theorem)

Accepted Answer

Relates the outward flux of a vector field $\mathbf{F}$ through a closed surface $S$ to the volume integral of its divergence over the enclosed region $E$:
$$\iint_S \mathbf{F} \cdot d\mathbf{S} = \iiint_E (\
abla \cdot \mathbf{F}) \, dV$$

Multivariable Calculus - Vector Calculus & Optimization

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