Differential vector operators

Question 1

Define the vector field ∇φ

Answer 1

∇φ is the gradient of φ. It always points toward the direction of maximum increase of φ

Question 2

Give the equation of the directional derivative of φ along u

Answer 2

α is the angle between u and ∇φ

|∇φ| is the magnitude of ∇φ

Question 3

Give the equation of the gradient of a rotationally symmetric scalar field φ(r) = f(r)

Answer 3

ê_r is the unit vector along the direction of r

Question 4

Define dr

Answer 4

Question 5

Give the equation of the total differential of a scalar field

φ(r) = φ(x, y, z)

Answer 5

Question 6

Give the equation of the total differential of a vector field A(r)

Answer 6

Question 7

Define the divergence of the vector field A(r)

Answer 7

Question 8

Define the curl of the vector field A(r)

Answer 8

Question 9

Define the Laplace operator

Answer 9

The Laplace operator applied to a scalar field φ is called the Laplacian