Lectures 4, 5 and 6

Question 1

How do you find the unit vector of something?

Answer 1

Unit vector = vector/|vector|

Question 2

What is the equation for ∇^2 in terms of d/dx and d/dy?

Answer 2

∇^2 = d^2/dx^2 + d^2/dy^2

Question 3

What is the equation for ∇^2 in terms of d/dρ and d/dθ?

Answer 3

∇^2 = d^2/dρ^2 + 1/ρ * d/dρ + 1/ρ^2 * d^2/dθ^2

Question 4

What is the equation for the Lagrange multiplier? What do the letters mean?

Answer 4

d(f+λg) = 0

where f is the original function, g is the constraint, and λ is the Lagrange multiplier.

Question 5

What is the total differential including the Lagrange multiplier?

Answer 5

d(f+λg) = (df/dx + λdg/dx)dx + (df/dy + λdg/dy)dy = 0

Question 6

What are the three simultaneous equations we can get from the total differential including the Lagrange multiplier?

Answer 6

The two coefficients of dx and dy, and the constraint g(x,y). df/dx + λ*dg/dx = 0, df/dy = λdg/dy = 0, and g(x,y) = c

Question 7

What is an important relationship including ∇f and ∇g and the lagrange multiplier?

Answer 7

∇f +λ∇g = 0

Question 8

What are the three basis vectors denoted by for cylindrical polar coordinates?

Answer 8

eρ, eФ and ez

Question 9

What are the equations for the three basis vectors for cylindrical polar coordinates?

Answer 9

eρ = cosФ i + sinФ j, 
eФ = -ρ*sinФ i + ρ*cosФ j
ez = k