If [math]f\left(x,y,z\right)[/math] is a function of three variables and the number [math]c[/math] is in the range of [math]f[/math], then the equation [math]f\left(x,y,z\right)=c[/math] defines a [i]level surface[/i] of [math]f[/math]. If the point [math]P_0\left(x_0,y_0,z_0\right)[/math] is on a level surface of [math]f[/math], then the gradient of [math]f[/math] at [math]P_0[/math], denoted [math]\nabla f|_{P_0}[/math], is perpendicular to that level surface. We can now find the plane tangent to the surface at [math]P_0[/math] by identifying the unique plane through [math]P_0[/math] that is normal to the vector [math]\nabla f|_{P_0}[/math].[br][br]In this interactive figure, define a function [math]f\left(x,y,z\right)[/math] and fix a level surface of that function. Then, place point [math]P_0[/math] on the surface and you can view the gradient vector and tangent plane at that point.

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