MATH201 - Multivariable Calculus & Vector Fields Flashcards

Question 1

What is a partial derivative?

Accepted Answer

A partial derivative of a multivariable function is the derivative with respect to one variable, treating all other variables as constants. For $f(x,y)$, $\frac{\partial f}{\partial x}$ is the derivative with respect to $x$ while holding $y$ constant.

Question 2

How do you calculate $\frac{\partial f}{\partial x}$ for $f(x,y) = x^2y + \sin(xy)$?

Accepted Answer

Treat $y$ as a constant.
$\frac{\partial f}{\partial x} = \frac{\partial}{\partial x}(x^2y) + \frac{\partial}{\partial x}(\sin(xy))$
$\frac{\partial f}{\partial x} = 2xy + \cos(xy) \cdot y$
$\frac{\partial f}{\partial x} = 2xy + y\cos(xy)$

Question 3

Define the gradient vector $
abla f$.

Accepted Answer

For a scalar function $f(x_1, x_2, \dots, x_n)$, the gradient vector $
abla f$ is a vector whose components are the partial derivatives of $f$ with respect to each variable.
For $f(x,y,z)$, $
abla f = \left\langle \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z} ightangle$.

Question 4

How do you find the gradient of $f(x,y,z) = x^2y + yz^3$?

Accepted Answer

$\frac{\partial f}{\partial x} = 2xy$
$\frac{\partial f}{\partial y} = x^2 + z^3$
$\frac{\partial f}{\partial z} = 3yz^2$
So, $
abla f = \langle 2xy, x^2 + z^3, 3yz^2 angle$.

Question 5

What is the geometric interpretation of the gradient vector $
abla f$?

Accepted Answer

The gradient vector $
abla f(P)$ points in the direction of the greatest rate of increase of $f$ at point $P$. Its magnitude, $|
abla f(P)|$, is this maximum rate of increase. It is also normal to the level curve/surface of $f$ at $P$.

Question 6

Define the directional derivative of $f$ in the direction of a unit vector $\mathbf{u}$.

Accepted Answer

The directional derivative $D_{\mathbf{u}}f(P)$ measures the rate of change of $f$ at point $P$ in the direction of the unit vector $\mathbf{u}$. It is defined as $D_{\mathbf{u}}f(P) = 
abla f(P) \cdot \mathbf{u}$.

Question 7

How do you calculate the directional derivative $D_{\mathbf{u}}f$?

Accepted Answer

First, calculate the gradient vector $
abla f$. Then, ensure $\mathbf{u}$ is a unit vector (normalize it if it's not). Finally, compute the dot product: $D_{\mathbf{u}}f = 
abla f \cdot \mathbf{u}$.

Question 8

In what direction does $f$ increase most rapidly, and what is that maximum rate?

Accepted Answer

The function $f$ increases most rapidly in the direction of the gradient vector $
abla f$. The maximum rate of increase is the magnitude of the gradient vector, $|
abla f|$.

Question 9

What does a double integral $\iint_R f(x,y)dA$ represent geometrically?

Accepted Answer

If $f(x,y) \ge 0$, the double integral $\iint_R f(x,y)dA$ represents the volume of the solid that lies above the region $R$ in the $xy$-plane and below the surface $z = f(x,y)$. More generally, it represents the signed volume.

Question 10

How do you set up a double integral over a rectangular region $R = [a,b] 	imes [c,d]$?

Accepted Answer

The double integral can be set up as an iterated integral:
$\iint_R f(x,y)dA = \int_a^b \int_c^d f(x,y) dy dx$ or
$\iint_R f(x,y)dA = \int_c^d \int_a^b f(x,y) dx dy$.
Fubini's Theorem guarantees these are equal if $f$ is continuous on $R$.

Question 11

What does a triple integral $\iiint_E f(x,y,z)dV$ represent?

Accepted Answer

If $f(x,y,z) = 1$, the triple integral $\iiint_E 1 dV$ represents the volume of the solid region $E$. More generally, it can represent mass if $f$ is a density function, or an average value of $f$ over $E$.

Question 12

What is the purpose of the method of Lagrange Multipliers?

Accepted Answer

The method of Lagrange Multipliers is used to find the local maximum and minimum values of a function $f(x,y,z)$ subject to a constraint $g(x,y,z) = k$.

Question 13

Outline the steps for finding extrema using Lagrange Multipliers.

Accepted Answer

To find the extrema of $f(x,y,z)$ subject to $g(x,y,z) = k$:
1. Solve the system of equations: $
abla f = \lambda 
abla g$ and $g(x,y,z) = k$.
2. The solutions $(x,y,z)$ are candidate points for extrema.
3. Evaluate $f$ at all candidate points to find the maximum and minimum values.

Question 14

Define the curl of a vector field $\mathbf{F} = \langle P, Q, R \rangle$.

Accepted Answer

The curl of a 3D vector field $\mathbf{F} = P\mathbf{i} + Q\mathbf{j} + R\mathbf{k}$ is a vector field defined as:
$	ext{curl } \mathbf{F} = 
abla 	imes \mathbf{F} = \left( \frac{\partial R}{\partial y} - \frac{\partial Q}{\partial z} ight)\mathbf{i} + \left( \frac{\partial P}{\partial z} - \frac{\partial R}{\partial x} ight)\mathbf{j} + \left( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} ight)\mathbf{k}$.

Question 15

What is the physical interpretation of the curl of a vector field?

Accepted Answer

The curl of a vector field measures the tendency of the field to rotate or "curl" around a point. If $	ext{curl } \mathbf{F} = \mathbf{0}$, the field is irrotational. Its direction indicates the axis of rotation, and its magnitude indicates the strength of the rotation.

Question 16

Define the divergence of a vector field $\mathbf{F} = \langle P, Q, R \rangle$.

Accepted Answer

The divergence of a 3D vector field $\mathbf{F} = P\mathbf{i} + Q\mathbf{j} + R\mathbf{k}$ is a scalar field defined as:
$	ext{div } \mathbf{F} = 
abla \cdot \mathbf{F} = \frac{\partial P}{\partial x} + \frac{\partial Q}{\partial y} + \frac{\partial R}{\partial z}$.

Question 17

What is the physical interpretation of the divergence of a vector field?

Accepted Answer

The divergence of a vector field measures the tendency of the field to expand or contract at a point. It represents the net outward flux per unit volume. If $	ext{div } \mathbf{F} = 0$, the field is incompressible or solenoidal.

Question 18

State Green's Theorem.

Accepted Answer

Green's Theorem relates a line integral around a simple closed curve $C$ in the plane to a double integral over the region $D$ bounded by $C$.
$\oint_C P\,dx + Q\,dy = \iint_D \left( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \right) dA$.
The curve $C$ must be traversed counterclockwise.

Question 19

When is Green's Theorem useful?

Accepted Answer

Green's Theorem is useful for:
1. Evaluating complicated line integrals by converting them to simpler double integrals.
2. Calculating the area of a region $D$ using a line integral: Area$(D) = \oint_C x\,dy = -\oint_C y\,dx = \frac{1}{2}\oint_C x\,dy - y\,dx$.
3. Determining if a 2D vector field is conservative.

Question 20

State Stokes' Theorem.

Accepted Answer

Stokes' Theorem relates a surface integral of the curl of a vector field over a surface $S$ to a line integral of the vector field around the boundary curve $C$ of $S$.
$\iint_S (	ext{curl } \mathbf{F}) \cdot d\mathbf{S} = \oint_C \mathbf{F} \cdot d\mathbf{r}$.
The orientation of $C$ must be consistent with the orientation of $S$ (right-hand rule).

Question 21

What is the relationship between Stokes' Theorem and curl?

Accepted Answer

Stokes' Theorem essentially states that the circulation of a vector field around a closed curve $C$ is equal to the total "curl" (or rotational tendency) of the field over any surface $S$ bounded by $C$. It provides a fundamental connection between line integrals and surface integrals involving curl.

Question 22

State the Divergence Theorem (Gauss's Theorem).

Accepted Answer

The Divergence Theorem relates the flux of a vector field $\mathbf{F}$ across a closed surface $S$ to the triple integral of the divergence of $\mathbf{F}$ over the solid region $E$ enclosed by $S$.
$\iint_S \mathbf{F} \cdot d\mathbf{S} = \iiint_E 	ext{div } \mathbf{F} \, dV$.
The surface $S$ must be oriented with outward-pointing normal vectors.

Question 23

What is a flux integral?

Accepted Answer

A flux integral (or surface integral of a vector field) measures the flow of a vector field $\mathbf{F}$ across a surface $S$. It quantifies how much of the field "passes through" the surface.

Question 24

How is the flux of a vector field $\mathbf{F}$ across a surface $S$ calculated?

Accepted Answer

The flux of $\mathbf{F}$ across an oriented surface $S$ with unit normal vector $\mathbf{n}$ is given by the surface integral:
$\iint_S \mathbf{F} \cdot d\mathbf{S} = \iint_S \mathbf{F} \cdot \mathbf{n} \, dS$.
If the surface is given by $\mathbf{r}(u,v)$, then $d\mathbf{S} = (\mathbf{r}_u 	imes \mathbf{r}_v) \, dA$, so $\iint_S \mathbf{F} \cdot (\mathbf{r}_u 	imes \mathbf{r}_v) \, dA$.

Question 25

How can you test if a 2D vector field $\mathbf{F} = \langle P, Q \rangle$ is conservative using partial derivatives?

Accepted Answer

A 2D vector field $\mathbf{F} = \langle P, Q angle$ is conservative if and only if $\frac{\partial P}{\partial y} = \frac{\partial Q}{\partial x}$ throughout a simply connected domain. If this condition holds, then $\mathbf{F}$ is the gradient of some scalar potential function $f$, i.e., $\mathbf{F} = 
abla f$.

MATH201 - Multivariable Calculus & Vector Fields

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