Calculus

Question 1

Define the derivative of a function, f(x)

Answer 1

df/dx = lim(h->0) [f(x+h)-f(x)]/h

Question 2

Define the partial derivative of a function f(x1, x2, …, xn)

Answer 2

df/dxi = lim(h->0) [f(x1, …, x(i-1), xi + h, x(i+1),…, xn) - f(x1, …, xn)]/h whenever the limit exists

Question 3

Define a smooth funtionn

Answer 3

A function that is differentiable for all orders

Question 4

Define the Laplacian of a function in two variables

Answer 4

The Laplacian of f(x,y) is defined by Δf = ∂^2f/∂x^2 + ∂^2f/∂y^2

Question 5

Define a harmonic function

Answer 5

f(x,y) is harmonic if Δf = 0

Question 6

Define the Jacobian matrix

Answer 6

If we are changing coordinate system from x and y to u and v, the Jacobian matrix is the 2x2 matrix:
∂x/∂u ∂x/∂v
∂y/∂u ∂y/∂v

Question 7

Define a Jacobian

Answer 7

The Jacobian of a coordinate change is the determinate of the Jacobian matrix describing the coordinate change.

Question 8

Give x and y in polar coordinates

Answer 8

x = rcosθ
y = rsinθ

Question 9

Give x and y in parabolic coordinates

Answer 9

x = 1/2(u^2 - v^2)
y = uv

Question 10

Give x, y and z in cylindrical coordinates

Answer 10

x = rcosθ
y = rsinθ
z = z

Question 11

Give x, y and z in spherical coordinates

Answer 11

x = rsinθcosφ
y = rsinθsinφ
z = rcosθ

Question 12

Define a tangent vector

Answer 12

Given a curve c(t) = (x(t),y(t)), the tangent vector is c’(t) = (x’(t),y’(t))

Question 13

Define the tangent line of a curve

Answer 13

Given a curve c(t) = (x(t),y(t)), the tangent line at c(t_0) is the line c(t_0) + λc’(t_0)

Question 14

Define the unit tangent vector

Answer 14

Given a curve c(t) = (x(t),y(t)), the unit tangent vector at c(t_0) is u(t_0) = c’(t_0)/|c’(t_0)|

Question 15

Define the unit normal vectors

Answer 15

The unit normal vectors are defined such that the angle between the unit tangent vector and n(t_0) is π/2 or -π/2 and |n(t_0)| = 1

Question 16

Define the tangent plane

Answer 16

The tangent plane at p = r(u_0,v_0) is the plane passing through p, parallel to ∂r/∂u(p) and ∂r/∂v(p)

Question 17

Give the equation of a paraboloid

Answer 17

z = x^2/a^2 + y^2/b^2

Question 18

Give the equation of an ellipsoid

Answer 18

x^2/a^2 + y^2/b^2 + z^2/c^2 = 1

Question 19

Give the equation of a 1-sheeted hyperloid

Answer 19

x^2/a^2 + y^2/b^2 - z^2/c^2 = 1

Question 20

Give the equation of a 2-sheeted hyperloid

Answer 20

x^2/a^2 - y^2/b^2 - z^2/c^2 = 1

Question 21

Give the equation of a hyperbolic paraboloid

Answer 21

z = x^2/a^2 - y^2/b^2

Question 22

Give the equation of a cone

Answer 22

z^2 = x^2 + y^2

Question 23

Define the gradient vector

Answer 23

∇f = ∂f/∂x1, … , ∂f/∂xn

Question 24

Define a conservative vector field

Answer 24

A vector field is conservative if the line integral depends only on the end points of C, but not C itself

Question 25

Define a directional derivative

Answer 25

If f: R^n -> R is a scalar function and u is a unit vector, the directional derivative of f in the direction u at the point a is:
∂uf(a) = lim(t->0) [f(a+tu) - f(a)]/t

Question 26

Define a level set

Answer 26

The set of points x where f(x) is equal to a given constant, ie. L_c = {x ∈ R^n, f(x) = c}

Question 27

Define a critical point

Answer 27

A point where ∇f = 0