MATH202 - Multivariable Calculus & Vector Field Theory Flashcards

Question 1

Gradient Vector ($
abla f$)

Accepted Answer

For a differentiable scalar function $f(x, y, z)$, the gradient vector is given by $
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 the steepest increase of $f$, and its magnitude equals the maximum rate of change.

Question 2

Directional Derivative

Accepted Answer

The rate of change of $f(x,y,z)$ in the direction of a unit vector $\vec{u} = \langle a, b, c angle$ is calculated using the dot product: $D_{\vec{u}} f = 
abla f \cdot \vec{u}$.

Question 3

Tangent Plane Equation

Accepted Answer

For a surface defined implicitly by $F(x,y,z) = k$, the tangent plane at point $P_0(x_0, y_0, z_0)$ is given by: $
abla F(x_0, y_0, z_0) \cdot \langle x - x_0, y - y_0, z - z_0 angle = 0$.

Question 4

Second Derivatives Test (2D Critical Points)

Accepted Answer

At critical point $(a,b)$ where $ abla f = \vec{0}$, evaluate $D = f_{xx}f_{yy} - (f_{xy})^2$: - $D > 0$ 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 5

Lagrange Multipliers

Accepted Answer

A method to optimize $f(x,y,z)$ subject to a constraint $g(x,y,z) = k$. Solve the system of equations: $
abla f = \lambda 
abla g$ along with $g(x,y,z) = k$, where $\lambda$ is the Lagrange multiplier.

Question 6

Jacobian Determinant (Change of Variables)

Accepted Answer

For transformation $x = x(u,v)$ and $y = y(u,v)$, the Jacobian $J(u,v) = \frac{\partial(x,y)}{\partial(u,v)} = \begin{vmatrix} \frac{\partial x}{\partial u} & \frac{\partial x}{\partial v} \ \frac{\partial y}{\partial u} & \frac{\partial y}{\partial v} \end{vmatrix}$. The area differential transforms as $dA = |J(u,v)| \, du \, dv$.

Question 7

Cylindrical Coordinates Transformation

Accepted Answer

Transformation equations:
$x = r \cos	heta$, $y = r \sin	heta$, $z = z$
Volume element: $dV = r \, dz \, dr \, d	heta$
Conversion: $r^2 = x^2 + y^2$, $	an	heta = \frac{y}{x}$.

Question 8

Spherical Coordinates Transformation

Accepted Answer

Transformation equations:
$x = ho \sin\phi \cos	heta$, $y = ho \sin\phi \sin	heta$, $z = ho \cos\phi$
Volume element: $dV = ho^2 \sin\phi \, dho \, d\phi \, d	heta$
Where $ho \ge 0$, $0 \le \phi \le \pi$, $0 \le 	heta < 2\pi$.

Question 9

Divergence of a Vector Field ($
abla \cdot \vec{F}$)

Accepted Answer

For $\vec{F} = \langle P, Q, R angle$, the divergence is a scalar defined as $
abla \cdot \vec{F} = \frac{\partial P}{\partial x} + \frac{\partial Q}{\partial y} + \frac{\partial R}{\partial z}$. It measures the net outward expansion/outflow rate at a point.

Question 10

Curl of a Vector Field ($
abla 	imes \vec{F}$)

Accepted Answer

For $\vec{F} = \langle P, Q, R angle$, curl is a vector defined as:
$
abla 	imes \vec{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} = \left\langle \frac{\partial R}{\partial y} - \frac{\partial Q}{\partial z}, \frac{\partial P}{\partial z} - \frac{\partial R}{\partial x}, \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} ightangle$.
It measures local rotation.

Question 11

Conservative Vector Field Conditions

Accepted Answer

A continuous vector field $\vec{F}$ is conservative if $\vec{F} = 
abla f$ for a scalar potential $f$. Equivalent properties on a simply connected domain:
1. $\oint_C \vec{F} \cdot d\vec{r} = 0$ for every closed loop $C$
2. $\int_C \vec{F} \cdot d\vec{r}$ is path independent
3. $
abla 	imes \vec{F} = \vec{0}$

Question 12

Fundamental Theorem for Line Integrals

Accepted Answer

If $\vec{F} = 
abla f$ is continuous and $C$ is a smooth curve parameterized by $\vec{r}(t)$ for $a \le t \le b$, then $\int_C 
abla f \cdot d\vec{r} = f(\vec{r}(b)) - f(\vec{r}(a))$.

Question 13

Green's Theorem (Standard Form)

Accepted Answer

Let $C$ be a positively oriented, piecewise-smooth, simple closed curve enclosing region $D$ in $\mathbb{R}^2$. Then:
$\oint_C (P \, dx + Q \, dy) = \iint_D \left( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \right) dA$.

Question 14

Green's Theorem Vector Forms

Accepted Answer

1. **Circulation-Curl Form:** $\oint_C \vec{F} \cdot d\vec{r} = \iint_D (
abla 	imes \vec{F}) \cdot \mathbf{k} \, dA$
2. **Flux-Divergence Form:** $\oint_C \vec{F} \cdot \mathbf{n} \, ds = \iint_D (
abla \cdot \vec{F}) \, dA$

Question 15

Parametric Surface Area Formula

Accepted Answer

For a surface $S$ parametrized by $\vec{r}(u,v)$ over a domain $D$, the total surface area is $A(S) = \iint_D |\mathbf{r}_u 	imes \mathbf{r}_v| \, du \, dv$, where $\mathbf{r}_u$ and $\mathbf{r}_v$ are partial tangent vectors.

Question 16

Surface Integral of a Scalar Function

Accepted Answer

The surface integral of $f(x,y,z)$ over parametrized surface $S = \vec{r}(u,v)$ is:
$\iint_S f(x,y,z) \, dS = \iint_D f(\vec{r}(u,v)) |\mathbf{r}_u 	imes \mathbf{r}_v| \, du \, dv$.

Question 17

Surface Integral of a Vector Field (Flux)

Accepted Answer

The flux of $\vec{F}$ across oriented surface $S = \vec{r}(u,v)$ is:
$\iint_S \vec{F} \cdot d\vec{S} = \iint_S (\vec{F} \cdot \mathbf{n}) \, dS = \iint_D \vec{F}(\vec{r}(u,v)) \cdot (\mathbf{r}_u 	imes \mathbf{r}_v) \, du \, dv$.

Question 18

Stokes' Theorem

Accepted Answer

Let $S$ be an oriented surface bounded by a simple, closed, smooth curve $C$ with compatible orientation. Then:
$\oint_C \vec{F} \cdot d\vec{r} = \iint_S (
abla 	imes \vec{F}) \cdot d\vec{S}$.

Question 19

Gauss's Divergence Theorem

Accepted Answer

Let $E$ be a simple solid region bounded by a closed surface $S$ with outward unit normal vector $\mathbf{n}$. Then:
$\iint_S \vec{F} \cdot d\vec{S} = \iiint_E (
abla \cdot \vec{F}) \, dV$.

Question 20

Laplacian Operator ($
abla^2 f$)

Accepted Answer

The Laplacian of scalar field $f$ is defined as the divergence of the gradient: $
abla^2 f = 
abla \cdot (
abla f) = \frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 f}{\partial y^2} + \frac{\partial^2 f}{\partial z^2}$. Functions satisfying $
abla^2 f = 0$ are called harmonic.

MATH202 - Multivariable Calculus & Vector Field Theory

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