Midterm 2 Study

Question 1

How can you prove that a vector field, F, is conservative?

Answer 1

Curl F = 0

Question 2

What is the formula for Curl

Answer 2

gradient cross F

Question 3

What is formula for Divergence for function F

Answer 3

Gradient dot F

Question 4

What does divergence tell us?

Answer 4

How much a vector field diverges, spreads out from a point

Question 5

What does curl tell us?

Answer 5

How much a vector field gyrates, and forms swirls (eddies)

Question 6

For a vector field, G, to exist such that curl G exits on the same plane, what needs to be satisfied?

Answer 6

Divergence ( Curl G) = 0
If the above function is not true, then vector field G does not exist.

Question 7

What is a parametric surface?

Answer 7

Like a parametric equation except each component is a function of 2 variables

Question 8

What do you need for the equation of a plane?

Answer 8

A normal vector and a point on the plane

Question 9

What is the equation for a plane

Answer 9

(Normal vector) dot (r-r0)=0
Where r= <x,y,z> (general point )
And r0 is known point on the plane

Question 10

How do you find the normal vector of a parametric surface at a point?

Answer 10

Find the tangent lines in both the u and v directions (where u and v are the variables used in the parametric surface equation) and cross product them. The tangent line in the u direction is the partial derivative of the parametric surface equation with respect to u.

Question 11

What is the formula for the area of a parametric surface?

Answer 11

Double integral over the domain S of the magnitude of the normal vector (Ru cross Rv)

Question 12

What is Greens theorem?

Answer 12

For a positively oriented curve that is closed and simply connected, the line integral of Pdx +Qdy = double integral over the domain of the closed loop of (dQ/dx -dP/dy) dA

Question 13

What’s an extension of Greens theorem when you want to take the line integral of a function over the tangential component of a line (Work) (F dot dr) AKA (Pdx+ Qdy)

Answer 13

It’s equal to the double integral over the domain D, area enclosed by curve, of curl F

Question 14

What’s an extension of greens theorem when you want to take a line integral of a function over a curve with respect to the normal of the curve (F dot dn)

Answer 14

= double integral over domain D, area enclosed by curve, of the divergence F

Question 15

To change variables when solving double integrals what does the inside of the integrals change to

Answer 15

F(x(u,v), y(u,v)) (jacobian) DA

Question 16

To change variables when solving double integrals what does the inside of the integrals change to

Answer 16

F(x(u,v), y(u,v)) (jacobian) DA

Question 17

What is a Jacobian

Answer 17

Cross product of dx/d(u,v) by dy/(d(u,v))

Question 18

Is F is a conservative vector field or can be written as..

Answer 18

The gradient of a potential function f

Question 19

What is the fundamental theorem of line integrals

Answer 19

If F is conservative, then the line integral of F(r(t)) is the f(r(b)) - f(r(a)) where f is F’s potential function and the bounds are t goes from a to b

Question 20

What is the parametric formula of a line? When given two points? r0 and r1

Answer 20

r(t) = (1-t)r0 + tr1