Directional Derivatives and Gradients

This set of flashcards covers key vocabulary related to directional derivatives and gradients in multivariable calculus.

Directional Derivative

The rate of change of a function in the direction of a unit vector.

Gradient

A vector that points in the direction of the maximum rate of change of a function.

Maximum Rate of Change

The directional derivative is maximized in the direction of the gradient vector.

Orthogonality of Gradient and Level Curves

The gradient vector is orthogonal to the level curves at a given point.

Partial Derivatives

Derivatives of a function with respect to one variable while holding others constant.

Unit Vector

A vector with a magnitude of 1, used to specify direction.

Computing Directional Derivatives

Use the formula: D_u f =
abla f ullet u, where u is the unit vector.

2D Directional Derivative Formula

Du f(x, y) = fx(x, y)a + f_y(x, y)b for unit vector u = egin{pmatrix} a \ b \ ext{,} \ ext{where} \ ||u|| = 1.

3D Directional Derivative Formula

Du f(x, y, z) = fx(x, y, z)a + fy(x, y, z)b + fz(x, y, z)c for unit vector u = egin{pmatrix} a \ b \ c \ ext{,} \ ext{where} \ ||u|| = 1.

Level Curve

A curve along which a function of two variables has a constant value.

Example Situation for Maximum Rate of Change

Finding the direction of the steepest ascent of a hill defined by a height function.

Negative Gradient

Points in the direction of the minimum rate of change of a function.