Sec 15.6 Directional Derivatives and the Gradient Vector

1. Sec 15.6 Directional Derivatives and the Gradient Vector Definition: Let f be a function of two variables. The directional derivative of f at in the direction of a unit vector is if this limit exists.

2. Theorem: If f is a differentiable function of x and y, then f has a directional derivative in the direction of any unit vector and Definition:The Gradient Vector If f is a function of x and y, then the gradient of f (denoted by gradf or ) is defined by Note:

3. Theorem: If f (x, y, z) is differentiable and , , then the directional derivative is Thegradient vector (gradf or ) is And

4. Maximizing the Directional Derivative Theorem: Suppose f is a differentiable function of two or three variables. The maximum value of the directional derivative is and it occurs when u has the same direction as the gradient vector .

5. Tangent Planes to Level Surfaces Definition The tangent plane to the level surface F(x, y, z) = k at the point is the plane that passes through P and has normal vector Theorem: The equation of the tangent plane to the level surface F(x, y, z) = k at the point is

6. Definition The normal line to a surface S at the point P is the line passing through P and perpendicular to the tangent plane. Its direction is the gradient vector at P. Theorem: The equation of the normal line to the level surface F(x, y, z) = k at the point is