MATH 200 Flashcards: Vectors, Partials and Key Formulas

Question 1

How do you find the angle between two vectors?

Accepted Answer

$\cos\theta = \dfrac{\mathbf a\cdot\mathbf b}{|\mathbf a|\,|\mathbf b|}$, so $\theta$ is the arccos of that fraction.
The angle is always between $0$ and $\pi$. A dot product of $0$ means the vectors are perpendicular.

Question 2

How do you compute the cross product of two vectors?

Accepted Answer

$\mathbf a\times\mathbf b = \langle a_2b_3 - a_3b_2,\ a_3b_1 - a_1b_3,\ a_1b_2 - a_2b_1\rangle$
The result is perpendicular to both $\mathbf a$ and $\mathbf b$. Its length is the area of the parallelogram with sides $\mathbf a$ and $\mathbf b$.

Question 3

What is the formula for the projection of $\mathbf b$ onto $\mathbf a$?

Accepted Answer

$\text{proj}_{\mathbf a}\mathbf b = \dfrac{\mathbf a\cdot\mathbf b}{|\mathbf a|^2}\,\mathbf a$
It points along $\mathbf a$. The component is $\dfrac{\mathbf a\cdot\mathbf b}{|\mathbf a|}$. And $\mathbf b - \text{proj}_{\mathbf a}\mathbf b$ is perpendicular to $\mathbf a$.

Question 4

How do you write the equation of a line in 3D?

Accepted Answer

$\langle x, y, z\rangle = \langle x_0, y_0, z_0\rangle + t\,\langle a, b, c\rangle$: a point on the line plus $t$ times a direction.
When $a, b, c \ne 0$, solve each component for $t$ to get the symmetric form $\dfrac{x - x_0}{a} = \dfrac{y - y_0}{b} = \dfrac{z - z_0}{c}$.

Question 5

What is the equation of a plane through a point with normal vector $\langle a, b, c\rangle$?

Accepted Answer

$a(x - x_0) + b(y - y_0) + c(z - z_0) = 0$
Going backwards: in $ax + by + cz = d$, the coefficients $\langle a, b, c\rangle$ form a normal vector.

Question 6

What is the formula for the distance from a point to a plane?

Accepted Answer

$D = \dfrac{|ax_0 + by_0 + cz_0 - d|}{\sqrt{a^2 + b^2 + c^2}}$ for the plane $ax + by + cz = d$.
Plug the point into $ax + by + cz - d$ and take the absolute value. Then divide by the length of the normal.

Question 7

What is the arc length formula for a curve $\mathbf r(t)$?

Accepted Answer

$L = \displaystyle\int_a^b |\mathbf r'(t)|\,dt$
$|\mathbf r'(t)|$ is the speed, so arc length is speed integrated over time.

Question 8

How do you tell a hyperboloid of one sheet from a hyperboloid of two sheets?

Accepted Answer

Write it with $+1$ on the right and count the minus signs on the left.
One minus: one sheet. Two minuses: two sheets.
Or slice across the axis. A slice at every height means one sheet. A gap with no slices means two.

Question 9

How do you show that a limit of a function of two variables does not exist?

Accepted Answer

Find two paths into the point that give different limits.
Paths that agree prove nothing. To show a limit does exist, use continuity, algebra, or polar coordinates and the squeeze theorem.

Question 10

What is the limit definition of the partial derivative $f_x(a, b)$?

Accepted Answer

$f_x(a, b) = \displaystyle\lim_{h \to 0}\dfrac{f(a + h, b) - f(a, b)}{h}$
Only $x$ moves. For $f_y(a, b)$, use $f(a, b + h)$ instead.

Question 11

When are the mixed partial derivatives $f_{xy}$ and $f_{yx}$ equal?

Accepted Answer

When both are continuous near the point. Then $f_{xy} = f_{yx}$. This is Clairaut's theorem.
So you can differentiate in whichever order is easiest. Also, when $P_y \ne Q_x$, no smooth $f$ has $f_x = P$ and $f_y = Q$.

Question 12

How do you find $\dfrac{\partial z}{\partial x}$ when $z$ is defined by $F(x, y, z) = 0$?

Accepted Answer

$\dfrac{\partial z}{\partial x} = -\dfrac{F_x}{F_z}$ and $\dfrac{\partial z}{\partial y} = -\dfrac{F_y}{F_z}$, where $F_z \ne 0$.
You get the same answer by differentiating both sides in $x$. Each time you differentiate a $z$, multiply by $\dfrac{\partial z}{\partial x}$.

Question 13

What is the chain rule for $w = f(x, y)$ when $x$ and $y$ depend on $t$?

Accepted Answer

$\dfrac{dw}{dt} = f_x\,\dfrac{dx}{dt} + f_y\,\dfrac{dy}{dt}$
Write one term per path in the tree diagram from $w$ down to $t$. Evaluate $f_x$ and $f_y$ at $(x(t), y(t))$.

Question 14

What is the equation of the tangent plane to $z = f(x, y)$ at $(a, b)$?

Accepted Answer

$z = f(a, b) + f_x(a, b)(x - a) + f_y(a, b)(y - b)$
The right side is also the linear approximation $L(x, y)$. The normal vector is $\langle f_x, f_y, -1\rangle$. It also gives the normal line.

Question 15

How do you find a normal vector to a surface $F(x, y, z) = c$?

Accepted Answer

Take the gradient at the point: $\mathbf n = \nabla F(x_0, y_0, z_0)$.
The tangent plane is then $\mathbf n \cdot \langle x - x_0, y - y_0, z - z_0\rangle = 0$.

Question 16

How do you estimate the maximum error in a calculated quantity?

Accepted Answer

Question 17

How do you find the directional derivative of a function?

Accepted Answer

$D_{\mathbf u}f = \nabla f \cdot \mathbf u$, where $\mathbf u$ must be a unit vector.

Question 18

In which direction does a function increase fastest, and how fast?

Accepted Answer

Along $\nabla f$, at rate $|\nabla f|$. It decreases fastest along $-\nabla f$, at rate $-|\nabla f|$.
Directions perpendicular to $\nabla f$ have rate $0$. They run along the level curve or surface.

Question 19

What is the second derivative test for $f(x, y)$?

Accepted Answer

$D = f_{xx}f_{yy} - f_{xy}^2$ at the critical point. $D > 0$, $f_{xx} > 0$: local min. $\ D > 0$, $f_{xx} < 0$: local max. $D < 0$: saddle. $\ D = 0$: no conclusion.

Question 20

How do you find the absolute max and min of $f(x, y)$ on a closed region?

Accepted Answer

Collect the critical points inside, the critical points on each boundary piece, and the corners. Evaluate $f$ at all of them and compare.
A continuous function on a closed, bounded region always has an absolute max and min. No second derivative test is needed.

Question 21

What equations do you solve in the method of Lagrange multipliers?

Accepted Answer

$\nabla f = \lambda\nabla g$ together with the constraint $g = k$.
At a max or min, the level curve of $f$ just touches the constraint curve. So the two gradients are parallel.

Question 22

How do you use Lagrange multipliers with two constraints?

Accepted Answer

Solve $\nabla f = \lambda\nabla g + \mu\nabla h$ with $g = k_1$ and $h = k_2$.
Five equations in $x, y, z, \lambda, \mu$. The candidates lie on the curve where the two surfaces meet.

Question 23

When should you reverse the order of integration in a double integral?

Accepted Answer

When the function in the inner integral has no elementary antiderivative, such as $e^{x^2}$, $\sin(y^2)$ or $\cos(x^3)$.
Redraw the region and slice it the other way. The new inner integral often gives the extra factor a substitution needs.

Question 24

What is dA in polar coordinates?

Accepted Answer

$dA = r\,dr\,d\theta$
Use it with $x = r\cos\theta$, $y = r\sin\theta$ and $x^2 + y^2 = r^2$. Forgetting the extra $r$ is the most common mistake.

Question 25

What is the circle $x^2 + y^2 = 2ax$ in polar coordinates?

Accepted Answer

$r = 2a\cos\theta$, for $-\tfrac{\pi}{2} \le \theta \le \tfrac{\pi}{2}$ (when $a > 0$).
Substitute $x^2 + y^2 = r^2$ and $x = r\cos\theta$, then divide by $r$. In the same way, for $a > 0$, $x^2 + y^2 = 2ay$ is $r = 2a\sin\theta$ for $0 \le \theta \le \pi$.

Question 26

How do you find the center of mass of a thin plate?

Accepted Answer

$\bar x = \dfrac{1}{M}\displaystyle\iint_D x\rho\,dA$ and $\bar y = \dfrac{1}{M}\displaystyle\iint_D y\rho\,dA$, where $M = \displaystyle\iint_D \rho\,dA$.
Here $\rho$ is the density. Each coordinate is a weighted average. With constant density, $\rho$ cancels and you divide by the area. That point is the centroid.

Question 27

What is the formula for the surface area of $z = f(x, y)$?

Accepted Answer

$S = \displaystyle\iint_D \sqrt{1 + f_x^2 + f_y^2}\,dA$, where $D$ is the surface's shadow in the $xy$-plane.
For a plane $ax + by + cz = d$ with $c > 0$, the root is the constant $\dfrac{\sqrt{a^2 + b^2 + c^2}}{c}$. So the surface area is that constant times the area of $D$.

Question 28

What is dV in cylindrical coordinates?

Accepted Answer

$dV = r\,dz\,dr\,d\theta$, with $x = r\cos\theta$, $y = r\sin\theta$, $z = z$.
The extra $r$ is the same one as in polar $dA = r\,dr\,d\theta$. Use cylindrical coordinates when the solid is symmetric around the $z$-axis.

Question 29

What is dV in spherical coordinates?

Accepted Answer

$dV = \rho^2\sin\varphi\,d\rho\,d\varphi\,d\theta$
A tiny box with sides $d\rho$, $\rho\,d\varphi$ and $\rho\sin\varphi\,d\theta$. Check: over the unit ball it gives $\tfrac{4\pi}{3}$.

Question 30

How do you convert spherical coordinates to Cartesian coordinates?

Accepted Answer

$x = \rho\sin\varphi\cos\theta$, $\ y = \rho\sin\varphi\sin\theta$, $\ z = \rho\cos\varphi$
$\rho$ is the distance to the origin. $\varphi$ is the angle down from the positive $z$-axis, with $0 \le \varphi \le \pi$. Going back: $\rho = \sqrt{x^2 + y^2 + z^2}$ and $\cos\varphi = \dfrac{z}{\rho}$.

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