MONASH MTH2010 - Multivariable Calculus & Vector Calculus Flashcards

Question 1

What is the partial derivative of $f(x,y)$ with respect to $x$?

Accepted Answer

The partial derivative $rac{\partial f}{\partial x}$ (or $f_x$) is the derivative of $f(x,y)$ with respect to $x$, treating $y$ as a constant. Mathematically, $rac{\partial f}{\partial x} = \lim_{h 	o 0} rac{f(x+h, y) - f(x,y)}{h}$.

Question 2

Explain $f_{xy}$ for a function $f(x,y)$.

Accepted Answer

$f_{xy}$ (or $rac{\partial^2 f}{\partial y \partial x}$) represents the second-order partial derivative where $f$ is first differentiated with respect to $x$, and then the result is differentiated with respect to $y$.

Question 3

State Clairaut's Theorem (or Schwarz's Theorem) regarding mixed partial derivatives.

Accepted Answer

If the second partial derivatives $f_{xy}$ and $f_{yx}$ are continuous on an open disk, then they are equal: $f_{xy}(x,y) = f_{yx}(x,y)$.

Question 4

State the Chain Rule for $z = f(x,y)$ where $x = g(t)$ and $y = h(t)$.

Accepted Answer

If $z = f(x,y)$ is a differentiable function of $x$ and $y$, and $x = g(t)$, $y = h(t)$ are differentiable functions of $t$, then $rac{dz}{dt} = rac{\partial z}{\partial x}rac{dx}{dt} + rac{\partial z}{\partial y}rac{dy}{dt}$.

Question 5

Define the gradient vector $
abla f$ for a scalar function $f(x,y,z)$.

Accepted Answer

The gradient vector $
abla f$ (read 'del f' or 'grad f') is a vector containing the partial derivatives of $f$: $
abla f(x,y,z) = \left\langle rac{\partial f}{\partial x}, rac{\partial f}{\partial y}, rac{\partial f}{\partial z} ightangle = rac{\partial f}{\partial x}\mathbf{i} + rac{\partial f}{\partial y}\mathbf{j} + rac{\partial f}{\partial z}\mathbf{k}$.

Question 6

What is the geometric significance of the gradient vector $
abla f(P)$ at a point $P$?

Accepted Answer

The gradient vector $
abla f(P)$ points in the direction of the maximum 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 surface $f(x,y,z) = k$ at $P$.

Question 7

How is the directional derivative $D_{\mathbf{u}} f$ defined, and what does it represent?

Accepted Answer

The directional derivative of $f$ in the direction of a unit vector $\mathbf{u}$ is $D_{\mathbf{u}} f = 
abla f \cdot \mathbf{u}$. It represents the rate of change of $f$ in the direction of $\mathbf{u}$.

Question 8

How do you find the equation of the tangent plane to a surface $F(x,y,z) = k$ at a point $(x_0, y_0, z_0)$?

Accepted Answer

The gradient vector $
abla F(x_0, y_0, z_0)$ is normal to the surface at $(x_0, y_0, z_0)$. The equation of the tangent plane is $rac{\partial F}{\partial x}(x_0, y_0, z_0)(x-x_0) + rac{\partial F}{\partial y}(x_0, y_0, z_0)(y-y_0) + rac{\partial F}{\partial z}(x_0, y_0, z_0)(z-z_0) = 0$.

Question 9

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

Accepted Answer

A double integral $\iint_R f(x,y) \,dA$ represents the volume of the solid bounded above by the surface $z=f(x,y)$ and below by the region $R$ in the $xy$-plane, assuming $f(x,y) \ge 0$. It is defined as a limit of Riemann sums.

Question 10

State Fubini's Theorem for evaluating double integrals over a rectangular region $R = [a,b] 	imes [c,d]$.

Accepted Answer

If $f$ is continuous on the rectangular region $R = [a,b] 	imes [c,d]$, then $\iint_R f(x,y) \,dA = \int_a^b \int_c^d f(x,y) \,dy \,dx = \int_c^d \int_a^b f(x,y) \,dx \,dy$.

Question 11

Why might you need to change the order of integration in a double integral, and how is it done?

Accepted Answer

Changing the order of integration (e.g., from $dy\,dx$ to $dx\,dy$) is often necessary when the original order leads to an intractable integral or when defining the region of integration is easier in the alternative order. It involves re-sketching the region and redefining the bounds.

Question 12

How do you transform a double integral $\iint_R f(x,y) \,dA$ into polar coordinates?

Accepted Answer

For a region $R$ suitable for polar coordinates, substitute $x = r\cos	heta$, $y = r\sin	heta$, and $dA = r\,dr\,d	heta$. The integral becomes $\iint_R f(r\cos	heta, r\sin	heta) \,r\,dr\,d	heta$. The Jacobian of the transformation is $r$.

Question 13

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

Accepted Answer

A triple integral $\iiint_E f(x,y,z) \,dV$ represents the integral of a function $f$ over a 3D solid region $E$. If $f(x,y,z)=1$, the integral gives the volume of $E$. It is defined as a limit of Riemann sums over small volume elements.

Question 14

How do you transform a triple integral $\iiint_E f(x,y,z) \,dV$ into cylindrical coordinates?

Accepted Answer

Substitute $x = r\cos	heta$, $y = r\sin	heta$, $z = z$, and $dV = r\,dz\,dr\,d	heta$. The integral becomes $\iiint_E f(r\cos	heta, r\sin	heta, z) \,r\,dz\,dr\,d	heta$. The Jacobian is $r$.

Question 15

How do you transform a triple integral $\iiint_E f(x,y,z) \,dV$ into spherical coordinates?

Accepted Answer

Substitute $x = ho\sin\phi\cos	heta$, $y = ho\sin\phi\sin	heta$, $z = ho\cos\phi$, and $dV = ho^2\sin\phi\,dho\,d\phi\,d	heta$. The integral becomes $\iiint_E f(ho\sin\phi\cos	heta, ho\sin\phi\sin	heta, ho\cos\phi) \,ho^2\sin\phi\,dho\,d\phi\,d	heta$. The Jacobian is $ho^2\sin\phi$.

Question 16

Define the line integral of a scalar function $f(x,y,z)$ along a curve $C$.

Accepted Answer

If $C$ is parameterized by $\mathbf{r}(t) = \langle x(t), y(t), z(t) angle$ for $a \le t \le b$, the line integral is $\int_C f(x,y,z) \,ds = \int_a^b f(x(t), y(t), z(t)) |\mathbf{r}'(t)| \,dt$, where $ds = |\mathbf{r}'(t)| \,dt$ is the arc length element.

Question 17

Define the line integral of a vector field $\mathbf{F}$ along a curve $C$.

Accepted Answer

The line integral of a vector field $\mathbf{F} = \langle P, Q, R angle$ along a curve $C$ parameterized by $\mathbf{r}(t)$ is $\int_C \mathbf{F} \cdot d\mathbf{r} = \int_a^b \mathbf{F}(\mathbf{r}(t)) \cdot \mathbf{r}'(t) \,dt$. This represents the work done by the force field $\mathbf{F}$ in moving a particle along $C$.

Question 18

State the Fundamental Theorem for Line Integrals.

Accepted Answer

If $\mathbf{F} = 
abla f$ is a conservative vector field (i.e., $\mathbf{F}$ is the gradient of some scalar function $f$, called a potential function), and $C$ is a smooth curve from $A$ to $B$, then $\int_C \mathbf{F} \cdot d\mathbf{r} = f(B) - f(A)$.

Question 19

State Green's Theorem (circulation form).

Accepted Answer

Let $C$ be a positively oriented, piecewise-smooth, simple closed curve in the plane, and let $D$ be the region bounded by $C$. If $\mathbf{F} = \langle P, Q angle$ is a vector field with continuous partial derivatives on $D$, then $\oint_C P\,dx + Q\,dy = \iint_D \left( rac{\partial Q}{\partial x} - rac{\partial P}{\partial y} ight) \,dA$.

Question 20

Define the curl of a 3D vector field $\mathbf{F} = \langle P, Q, R angle$.

Accepted Answer

The curl of $\mathbf{F}$ is a vector field defined as $	ext{curl } \mathbf{F} = 
abla 	imes \mathbf{F} = \left\langle rac{\partial R}{\partial y} - rac{\partial Q}{\partial z}, rac{\partial P}{\partial z} - rac{\partial R}{\partial x}, rac{\partial Q}{\partial x} - rac{\partial P}{\partial y} ightangle$. It measures the tendency of the field to rotate a small object.

Question 21

Define the divergence of a 3D vector field $\mathbf{F} = \langle P, Q, R angle$.

Accepted Answer

The divergence of $\mathbf{F}$ is a scalar field defined as $	ext{div } \mathbf{F} = 
abla \cdot \mathbf{F} = rac{\partial P}{\partial x} + rac{\partial Q}{\partial y} + rac{\partial R}{\partial z}$. It measures the outward flux per unit volume at a point, indicating whether the point is a source or a sink.

Question 22

State Stokes' Theorem.

Accepted Answer

Let $S$ be an oriented, piecewise-smooth surface bounded by a simple, closed, piecewise-smooth curve $C$ with positive orientation. If $\mathbf{F}$ is a vector field whose components have continuous partial derivatives on an open region containing $S$, then $\oint_C \mathbf{F} \cdot d\mathbf{r} = \iint_S (
abla 	imes \mathbf{F}) \cdot d\mathbf{S}$.

Question 23

State the Divergence Theorem (Gauss's Theorem).

Accepted Answer

Let $E$ be a simple solid region bounded by a closed surface $S$ with positive (outward) orientation. If $\mathbf{F}$ is a vector field whose components have continuous partial derivatives on an open region containing $E$, then $\iint_S \mathbf{F} \cdot d\mathbf{S} = \iiint_E (
abla \cdot \mathbf{F}) \,dV$.

Question 24

Define the surface integral of a scalar function $f(x,y,z)$ over a surface $S$.

Accepted Answer

If $S$ is parameterized by $\mathbf{r}(u,v)$ over a region $D$, the surface integral is $\iint_S f(x,y,z) \,dS = \iint_D f(\mathbf{r}(u,v)) |\mathbf{r}_u 	imes \mathbf{r}_v| \,dA$, where $dS = |\mathbf{r}_u 	imes \mathbf{r}_v| \,dA$ is the surface area element.

Question 25

Define the surface integral of a vector field $\mathbf{F}$ over an oriented surface $S$ (flux integral).

Accepted Answer

The flux of $\mathbf{F}$ across $S$ is $\iint_S \mathbf{F} \cdot d\mathbf{S} = \iint_S \mathbf{F} \cdot \mathbf{n} \,dS$, where $\mathbf{n}$ is the unit normal vector to $S$. If $S$ is parameterized by $\mathbf{r}(u,v)$, then $d\mathbf{S} = (\mathbf{r}_u 	imes \mathbf{r}_v) \,dA$, so $\iint_D \mathbf{F}(\mathbf{r}(u,v)) \cdot (\mathbf{r}_u 	imes \mathbf{r}_v) \,dA$.

MONASH MTH2010 - Multivariable Calculus & Vector Calculus

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