Directional Derivatives

Question 1

What is the directional derivative?

Answer 1

DD of f at point a,b in the direction of the unit vector u = (x0, y0) is given by the limit in notes.

If we have a multivariate function, the DD at the point a,b, which is a point in the domain of f = rate of change of z in the direction of unit vector u.

The point P(a, b, c) will lie on surface S. The vertical plane passing through P in the direction of the unit vector, intersects S in a Curve C.

The tangent line T to C at P, is the rate of change of z in the direction of u.

Question 1

What is the relationship between Directional Derivative and Partial Derivative?

Answer 1

If u = 1, 0
then directional derivative is equal to partial derivative of x.
same for y.

Question 2

What is the length of the direction vector?

Answer 2

Unit vector, so u = v / |v|

Question 3

What is the formula for the directional derivative?

Answer 3

Df(x,y) = fx(x, y)a + fy(x, y)b

Question 5

What is the gradient vector of f, and how is it connected to DD?

Answer 5

Points in direction of steepest inclinde at the point, and has a magnitude

GV f = vector defined by partial derivatives x, y

Since DD = fx(x, y)a + fy(x, y)b,
(fx, fy) x (a,b)
= GV * u

Question 6

When is the DD 0?

Answer 6

When u and GV are orthogonal (dot product = 0)