Suppose [math]f\left(x,y\right)[/math] is a real-valued function of two variables. The [i]gradient [/i]of [math]f[/math] is the vector-valued function denoted [math]\nabla f[/math] and defined by[br] [math]\nabla f=f_x\left(x,y\right)\mathbf{i}+f_y\left(x,y\right)\mathbf{j}[/math].[br][br]If [math]\left(x_0,y_0\right)[/math] is a point in the domain of [math]f[/math], then the [i]gradient vector of [/i][math]f[/math][i] at the point[/i] [math]\left(x_0,y_0\right)[/math] is the vector [br] [math]\left(\nabla f\right)_{\left(x_0,y_0\right)}=f_x\left(x_0,y_0\right)\mathbf{i}+f_y\left(x_0,y_0\right)\mathbf{j}[/math].[br][br]The gradient vector has two important properties:[br] 1) It points in the direction of greatest increase of [math]f[/math], and[br] 2) Its magnitude is the rate of change of [math]f[/math] in that direction.[br][br]By point (1), the gradient vector at [math]\left(x_0,y_0\right)[/math] must be perpendicular to any level curve passing through [math]\left(x_0,y_0\right)[/math]. In the figure, create [math]f\left(x,y\right)[/math] and a sequence of level curves. Then observe the behavior of the gradient vector at various points; see how this behavior relates to the underlying level curves. By clicking the checkbox to show the gradient field, you can see (scaled) gradient vectors at various points throughout the plane.

[i]Developed for use with Thomas' Calculus, published by Pearson.[/i]