Section 15.6 Directional Derivatives and the Gradient Vector.

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Presentation transcript:

Section 15.6 Directional Derivatives and the Gradient Vector

The directional derivative of f at (x 0, y 0 ) in the direction of a unit vector is if this limit exists. THE DIRECTIONAL DERIVATIVE

COMMENTS ON THE DIRECTIONAL DERIVATIVE If u = i, then D i f = f x. If u = j, then D j f = f y.

THE DIRECTIONAL DERIVATIVE AND PARTIAL DERIVATIVES 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 D u f (x, y) = f x (x, y) a + f y (x, y) b

THE DIRECTIONAL DERIVATIVE AND ANGLES If the unit vector u makes an angle θ with the positive x-axis, then we can write and the formula for the directional derivative becomes D u f (x, y) = f x (x, y) cos θ + f y (x, y) sin θ

VECTOR NOTATION FOR THE DIRECTIONAL DERIVATIVE

THE GRADIENT VECTOR If f is a function of two variables x and y, then the gradient of f is the vector function defined by NOTATION: Another notation for the gradient is grad f.

THE DIRECTIONAL DERIVATIVE AND THE GRADIENT The directional derivative can be expressed by using the gradient

The directional derivative of f at (x 0, y 0, z 0 ) in the direction of a unit vector is if this limit exists. THE DIRECTIONAL DERIVATIVE IN THREE VARIABLES

VECTOR FORM OF THE DEFINITION OF THE DIRECTIONAL DERIVATIVE The directional derivative of f at the vector x 0 in the direction of the unit vector u is NOTE: This formula is valid for any number of dimensions: 2, 3, or more.

THE GRADIENT IN THREE VARIABLES If f is a function of three variables, the gradient vector is The directional directive can be expressed in terms of the gradient as

Theorem: Suppose f is a differentiable function of two or three variables. The maximum value of the directional derivative D u f (x) is and it occurs when u has the same direction as the gradient vector. MAXIMIZING THE DIRECTIONAL DERIVATIVE

Let F be a function of three variables. The tangent plane to the level surface F(x, y, z) = k at P(x 0, y 0, z 0 ) is the plane that passes through P and is tangent to F(x, y, z) = k. Its normal vector is, and its equation is TANGENT PLANE AND NORMAL LINE TO A LEVEL SURFACE The normal line to the level surface F(x, y, z) = k at P is the line passing through P and perpendicular to the tangent plane. Its direction is given by the gradient, and its symmetric equations are