Functions of Several Variables

Question 1

What is a Directional Derivative?

Answer 1

Rate of change of f(x,y,z) in arbitrary direction u = (u,v,w)

Different for each point on the graph.

Question 2

The Gradient Vector

Answer 2

∇f = (∂f/∂x , ∂f/∂y , ∂f/∂z)

Question 3

Directional Derivative Formula

Answer 3

u/mag(u) * ∇f

Question 4

Steps of Directional Derivative Questions

Question 5

Magnitude of Directional Derivative

Answer 5

mag(∇f)

Question 6

FoSV Chain Rule

Answer 6

df/dt = (∂f/∂x)(∂x/∂t) + (∂f/∂y)(∂y/∂t)

Question 7

Hessian Matrix

Answer 7

Matrix of second partial derivatives for a 2 variable function.

H(f) = ( fxx fxy )
( fyx fyy )

Question 8

What is Hessian Matrix used for

Answer 8

Used to help identify stationary points of a 2 variable function.

Question 9

How to find stationary points of a 2 variable function?

Answer 9

1) Calculate fx = 0 & fy = 0
2) Calculate Determinant of Hessian Matrix @ fx & fy =0
3) Calculate fxx

Question 10

Seperation of Variables for PDEs

Question 11

Multiple Integration

Answer 11

∫∫ f(x)*g(y) dxdy

1) Integrate w.r.t dx
2) Integrate w.r.t dy

Doesn’t matter which order they are in.

Question 12

Using Polar Co-ordinates in Multiple Integrals

Answer 12

x = rcos(θ) y = rsin(θ)
where:
r = Magnitude
θ = Argument (like complex numbers)

Do straight replacement and use Jacobian Matrix.