Multivariable Calculus

Question 1

Define the centre of mass of a surface

Answer 1

Centre of mass = 1/M * Triple integral of rρ(r)dV across the entire surface, where ρ(r) if a function of the density of the surface.

Question 2

Define a scalar field on R^3

Answer 2

A map from R^3 to R

Question 3

Define a vector field on R^3

Answer 3

A map from R^3 to R^3

Question 4

Define the moment of inertia of a region in the plane

Answer 4

The double integral over the region of
ρ(x, y)[(x − x0)^2 + (y − y0)^2]dA, where ρ is the density per unit area and (x0,y0) is the point we are taking the moment of inertia about.

Question 5

Define the Jacobian of a coordinate change.

Answer 5

The determinant of the matrix:
∂u/∂x ∂u/∂y ∂u/∂z
∂v/∂x ∂v/∂y ∂v/∂z
∂w/∂x ∂w/∂y ∂w/∂z

Question 6

Define the median of a function on a 3D region

Answer 6

The value of m that satisfies:
Vol ({(x, y, z):f(x, y, z) ≤ m}) = Vol(R)/2

Question 7

Define a planar angle

Answer 7

The planar angle subtended at O by two lines is the arc length of the circle of radius r divided by its radius (in radians).

Question 8

Define a solid angle

Answer 8

Surface area on sphere/Radius^2 (in steridians).

Question 9

Define a curve

Answer 9

A piecewise smooth function mapping from an interval to R^3.

Question 10

Define a simple curve

Answer 10

If γ:[a,b] -> R^3, γ is simple if γ is 1-1 with the one exception that γ(a) = γ(b) is permitted.

Question 11

Define a closed curve

Answer 11

If γ: [a,b] -> R^3 and γ(a) = γ(b), then the curve is closed

Question 12

Define the arc length of a curve

Answer 12

If C is parameterised by t, the arc length is the integral of |γ’(t)|dt.

Question 13

Define a conservative field

Answer 13

A vector field F: S->R^3 is conservative if there exists a scalar field φ: S -> R such that F = ∇φ

Question 14

Define a potential

Answer 14

if F = ∇φ, then φ is a potential for F.

Question 15

Define path connected

Answer 15

S ⊆ R^3 is path connected if for all p and q in S, there exists γ:[a,b] -> S such that γ(a) = p and γ(b) = q.

Question 16

Define grad

Answer 16

The gradient of a scalar field φ is ∇φ

Question 17

Define curl

Answer 17

The curl of a vector field F is ∇^F

Question 18

Define the Laplacian

Answer 18

The Laplacian of a scalar field φ is ∇.∇φ, while the Laplacian of a vector field F is ∇.∇F.

Question 19

Define div

Answer 19

The divergence of a vector field F is ∇.F

Question 20

Define a convex region

Answer 20

A region R is convex if for any p, q in R, the line segment connecting p and q is contained in R.