The Gradient of a Scalar Field

Question 1

Define ∇, del.

Answer 1

Question 2

What is the symbol for del.

Answer 2

∇

Question 3

What does ∇ stand for?

Answer 3

del or nabla

Question 4

Define _∇_f

Answer 4

Question 5

What is the symbol for gradient or grad f?

Answer 5

_∇_f

Question 6

What does _∇_f stand for?

Answer 6

Gradient or grad f

Question 7

What components does the vector field grad f have?

Answer 7

Question 8

In what direction does _∇_f go for lines of constant f?

Answer 8

Normal to them

Question 9

What are three properties of scalar fields?

Answer 9

Question 10

Prove the following when n=2.

Answer 10

Question 11

Prove the following when n=2.

Answer 11

Question 12

Prove the following in n dimensions.

Answer 12

Question 13

If we use arc length s as the parameter then how does the following change?

Answer 13

Question 14

Define the directional derivative of f in the direction ñ.

Answer 14

Question 15

In which direction does _∇_f point in?

Answer 15

The direction where f increases fastest.

Question 16

Prove that _∇_f points in the direction where f increases fastest.

Answer 16

Question 17

What is a tangent plane?

Answer 17

In ℝ³ the plane through P orthogonal to _∇_f is called the tangent plane to S at P. The tangent to any curve in S through P lies in this plane.