Final Exam

Question 1

What are three methods to solving limit problems?

Answer 1

1. Try along different paths, such as (x,0), (0,y), (x,x), (y^2, y)…
2. Convert to polar coordinates.
3. Use the Squeeze theorem. This can work for trig functions or e; e^-1 < e^(y^2) < e.

Question 2

How do you find the equation of a plane?

Answer 2

• Find the normal vector to the plane.
• Make these components the coefficients in the equation.
• Pick some point on the plane.
• (n)(x - x0) + n(y - y0) = 0

Question 3

How do you find the normal vector to a plane?

Answer 3

• Given that <0,0,1>, <1,2,0>, <1,1,1> lie in the plane.
• Subtract the first from the other two.
• = <1,2,-1> , <1,1,0>
• Compute the cross product.
• <1,2,-1>
• <1,1, 0>
• = <1, -1, 1>

Question 4

How would you find the directional derivative of f?

Answer 4

Dot the gradient with the unit vector in the direction you want.

If you want to go in the maximum rate of change, dot the gradient with itself.

Question 5

What is the vector equation of a line?

Answer 5

r(t) = (starting point) + tv, where v is a parallel vector

Question 6

Compute the equation of the plane given a point and a line.

Answer 6

Use the vector from the line as one term in the cross product.

Use the vector from subracting (given point) - (equation point) as the other term.

Question 7

What are cylindrical coordinate substitutions?

Answer 7

Cylindrical Coordinates
Jacobian = dV = r [dz dr dt]
x = rcos(t)
y = rsin(t)
z = z

Question 8

What are spherical coordinate substitutions?

Answer 8

Spherical Coordinates
Jacobian = dV = r^2 sin(ø) [dr, dt, dø]
x = r cost sinø
y = r sint sin ø
z = r cos ø
r^2 = x^2 + y^2 + z^2

Question 9

How do you compute the arc length of a function?

Answer 9

Apply ∫ from a to b for the absolute value of r’(t).
∫ |r’(t) | dt
How to find the boundaries for t? Sometimes, the given vector function will have a straight “t” which one can solve for?

Question 10

For any vector v, we have v x v =

Answer 10

0

Question 11

If f is a function of three variables that has continuous second order partial derivatives, then _____ = 0.

Answer 11

curl(gradient • F) = 0

Question 12

Iff a vector field F is conservative, then its curl is

Answer 12

zero.

Question 13

If you take the divergence of a curl of the vector field, you will get ___.

Answer 13

0.

Question 14

Given that r”(t) ≠ a constant, is it true that r’(t) x r”(t) ≠ constant?

Answer 14

Plug in an example r(t).
For example, r(t) = <t^3, 0, 0>
Then r’(t) = <3t^2, 0,0> , r”(t) = <6t, 0, 0>, and their curl = 0.
0 is a constant, so this is false.

Question 15

How do you apply the FTLI?

Answer 15

Check that Py = Qx. (This shows that it is conservative.)
Construct some function f such that the gradient of f = F.
Take the partials of F with respect to x and y.
Integrate one.
Solve for any differences.
Take f(endpoint) - f(starting point).

Question 16

Given that r”(t) ≠ a constant, can r’(t) • r’(t) be constant?

Answer 16

Clever. Consider

Sin and Cos.

Take r(t) = <cos(t), sin(t), 0>.

Then r’(t) = < -sin(t), cos(t), 0> dotting with

With r’(t) = < -sin(t), cos(t), 0>

= sin^2(t) + cos^2(t) = 1

Question 17

If asked if some manipulation of theorems is true, what’s the first thing you should do?

Answer 17

Check to see whether they would result in scalars / vectors.

Dot product = scalar

Cross product = vector

Question 18

The value of the line integral does not depend on the parametrization of the curve, provided that ___

Answer 18

the curve is traversed exactly onece as t increases from a to b.

Question 19

What is the “jacobian” of ds?

Answer 19

ds = sqrt( (dx/dt)^2 + (dy/dt)^2) dt

Question 20

How do you find the line integral with respect to x?

Answer 20

∫ f(x,y) dx = f( x(t), y(t) ) , x’(t) dt

Question 21

What’s a vector way to solve this line integral

∫F • dr =

Answer 21

From a to b of F(r(t) • r’(t) dt

You’re plugging in the vector parametrization to the vector field, and then dotting with the derivative of the vector curve. Everything’s in “t.”

Question 22

By the Divergence Theorem, ∫∫{S} F • ds =

Answer 22

∫∫∫ {E} div(F)

Question 23

By the Divergence theorem, ∫∫∫ {E} div(F) =

Answer 23

∫∫F•ds over the boundary S

Question 24

How do you find the surface integral ∫∫{S} F•dS?

Answer 24

Normally, find a parametric representation of the surface. Then take the partials and take the cross product.

Then take ∫∫ {D} F(r) | r x r | dA

Question 25

What are the bounds for z in spherical coordinates?

Answer 25

From 0 to π

Question 26

Say you want to close a hemisphere. How would you do so?

Answer 26

The hemisphere consists of the hemisphere minus the unit disk boundary. Solve for the entire unit sphere using spherical coordinates, and then subtract – really, add, since its normal vector is downards – the unit disk.

When taking the surface integral of the unit disk, its normal vector is -k, and z = 0. This simplifies the calculation considerably.

Question 27

What does Stokes Theorem say?

Answer 27

∫∫{S} curl F • dS = ∫{C} F • dr

Question 28

Via Stokes,

∫{C} F • dr =

Answer 28

= ∫∫{S} curl F • dS

Question 29

Explain the three ways of directly evaluating surface integrals.

Answer 29

Method 1: If you have some function z, then then take ∫∫ (given) sqrt (dz/dx + dy + 1)

Convert to polar to solve.

Method 2: If you have some spherical function, parametrize it using spherical coordinates, and then take ∫∫ F(r) r(theta) x rø d(theta) dø. That is, cross the partials of the parametrization.

Method 3: If you have some z, then find a normal vector and dot it with the function F. A good normal vector is just the gradient of the boundary function.

Again, you’ll convert to polar to solve.

Question 30

How can you show that F is conservative?

Answer 30

Method 1: Compare it with Pi + Qj. If the partial of P wrt y (eg PAY) is equal to the partial of Q wrt x (QUIX?), then the vector field is conservative if the doman of F is open and simply connected.

Method 2: Show that there exists a function f such that ∆f = F. (where ∆ is actually del).

Method 3: Show that the curl = 0.

Question 31

You’re creating a potential function. You’ve already taken the integral of fx (i.e., the i term) and it has some constant of integration in it like g(x,z). What do you do?

Answer 31

Differentiate with respect to y.

Compare this new term with fx (i.e., the j term). Set them equal to solve for the g’(x,y) term.

Question 32

State Green’s Theorem.

Answer 32

∫∫{D} (Q wrt y) - ( P wrt y) = ∫ Pdx + Qdy.

That is, if you have some line integral that’s in PADDQ form, you can take the (?) integral of the area rather than a line integral if you take it with respect to QUIX MINUS PAY.

Question 33

What is the second derivative test for critical points?

Answer 33

Multiply two foxes by two eyes. Subtract the square of a mixed second derivative.

fxx*fyy - (fxy)2

If > 0 and fxx > 0, then a minimum.

If > 0 and fxx < 0, then a maximum.

If D < 0, then it is a saddle point.