Gradients

Gradients The partial derivatives of a function of several variables can be neatly packaged into a vector. In two variables

Definition: the Gradient of a real-valued function $z = f(x, \,y)$ is the vector function $(\nabla f)(x, \,y)\ = \ \frac{\partial f}{\partial x}\, {\mathbf i} \, +\, \frac{\partial f}{\partial y}\, {\mathbf j}$ whose components are the partial derivatives of $f$ in the $x$ - and $y$ -directions.

This definition generalizes immediately to a function

$w = f(x,\,y,\, z)$ :

$(\nabla f)(x,\,y,\, z) \ = \ \frac{\partial f}{\partial x}\, {\mathbf i} + \frac{\partial f}{\partial y}\, {\mathbf j} + \frac{\partial f}{\partial z}\, {\mathbf k}\,.$

Example: for

$f(x,\,y)= x^2 - y^2$ ,

$(\nabla f)(x,\, y) \ = \ 2x\,{\mathbf i} -2y\,{\mathbf j}\,.$ To draw the graph of

$\nabla f$ we select a set of points

$P(x,\,y)$ and represent

$(\nabla f)(P)$ by a vector with initial point

$P$ and length scaled so that it's not too long but remains a fixed proportion of the true length of

$(\nabla f)(P)$ . Drawing

$(\nabla f)(P)$ for every

$P$ is not helpful, of course, because it would cover the plane with arrows, so we choose a representative set. But for functions of

$3$ or more variables, the graph of

$\nabla f$ is usually too cluttered to be of much use even if a computer graphics program is used.

Many important properties of the gradient of

$z = f(x,\,y)$ are suggested by superimposing the graph of

$(\nabla f)(x,\,y)$ on the contour map of

$z = f(x,\,y)$ :

We'll explore these properties more in the next slide.

Chapter 10: Parametric Equations and Polar Coordinates

Chapter 12: Vectors and the Geometry of Space

Chapter 13: Vector Functions

Chapter 14: Partial Derivatives

Learning module LM 14.1: Functions of 2 or 3 variables:

Learning module LM 14.3: Partial derivatives:

Learning module LM 14.4: Tangent planes and linear approximations:

Learning module LM 14.5: Differentiability and the chain rule:

Learning module LM 14.6: Gradients and directional derivatives:

Learning module LM 14.7: Local maxima and minima:

Learning module LM 14.8: Absolute maxima and Lagrange multipliers:

Chapter 15: Multiple Integrals

Gradients