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]In the accompanying interactive figure, move the red point to set a point in the domain of [math]f[/math]. The gradient vector is drawn and you can confirm the two properties above. Note that while the surface is 3-dimensional, the gradient vector lies in the [math]xy[/math]-plane.