Gradients and directional derivatives

Definition 1. Let z=f(x, y), the gradient of f, denoted by , is defined by the vector

If w= f(x, y, z), the gradient of f denoted by , is defined by

Note: the gradient of a function is a vector.

Definition of directional derivative. If w=f(x, y, z) and (unit vector), then the directional derivative of w in the direction of is defined by

provided the limit exists.

Theorem 1. If z=f(x, y), and if are both continuous, and is a unit vector in the direction of then

(dot product of the gradient and the unit vector gives the directional derivative in the direction of a given vector )

Theorem 2. If w=f(x, y, z), and if are continuous, and , then

(dot product of the gradient and the unit vector gives the directional derivative in the direction of the given vector)

The vector Chain rule:

is a differentiable function of t, then

.

Definition 2. is called the normal derivative because it is normal to the level curve and it is denoted by .

3.Differentiation of composite and implicit functions. Tangent and surface normal

The case of one independent variable

Theorem 3. If is differentiable function in point and , are differentiable functions of independent variable , then derivative of composite function calculated by the formula:

.

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