Multivariable Calculus: Vector Calculus flashcards

Question 1

What is a vector field in ℝ²?

Accepted Answer

A rule giving a vector F(x, y) at every point (x, y) of a set D in ℝ²

Question 2

When is a vector field F called conservative?

Accepted Answer

When there is a scalar function f with ∇f = F

Question 3

What is a potential function for a vector field F?

Accepted Answer

A scalar function f such that ∇f = F

Question 4

What is a line integral?

Accepted Answer

An integral of a function taken along a plane or space curve

Question 5

What is a simple curve?

Accepted Answer

A curve that does not cross itself

Question 6

What is a closed curve?

Accepted Answer

A curve whose starting point and ending point coincide

Question 7

What does the Fundamental Theorem for Line Integrals state?

Accepted Answer

∫_C ∇f · dr = f(r(b)) − f(r(a)): the integral uses only the values of f at the ends of C

Question 8

What does it mean for a vector field F to be independent of path?

Accepted Answer

∫ F · dr is the same over any two curves in its domain with the same initial and terminal points

Question 9

What is the circulation of a conservative field on a simply connected domain around a closed curve?

Accepted Answer

Zero

Question 10

A field F has the cross-partial property on a simply connected domain. What follows?

Accepted Answer

F is conservative (it has a potential function)

Question 11

What are the two forms of Green's theorem?

Accepted Answer

The circulation form (integrand F · T) and the flux form (integrand F · N)

Question 12

For a source-free field F = ⟨P, Q⟩, what is a stream function g?

Accepted Answer

A function with P = g_y and Q = −g_x

Question 13

What is the curl of F = ⟨P, Q, R⟩?

Accepted Answer

∇ × F = (R_y − Q_z) i + (P_z − R_x) j + (Q_x − P_y) k

Question 14

Is the divergence of a vector field a scalar function or a vector field?

Accepted Answer

A scalar function

Question 15

On a simply connected domain, what curl condition makes a vector field conservative?

Accepted Answer

Its curl must be zero (and zero curl suffices)

Question 16

How many parameters are needed, in general, to parameterize a surface?

Accepted Answer

Two

Question 17

Name a surface that cannot be oriented.

Accepted Answer

A Möbius strip

Question 18

What surface integral gives the area of a surface S?

Accepted Answer

∬_S dS

Question 19

When is a parameterization r(u, v) regular?

Accepted Answer

When r_u × r_v is never zero in the parameter domain

Question 20

What does Stokes' theorem relate?

Accepted Answer

A flux integral over a surface S and a line integral around S's boundary curve C

Question 21

What does the divergence theorem relate?

Accepted Answer

A surface integral over a closed surface S and a triple integral over the solid inside S

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