Gradient

The gradient appears when the differential is written as a dot product: \(df=\nabla f\cdot\langle dx,dy\rangle\). Hence, for a unit direction \(\mathbf{u}\), the rate of change is \(\nabla f\cdot\mathbf{u}=|\nabla f|\cos\theta\), with \(\theta\) the angle between them. It is maximal, equal to \(|\nabla f|\), when \(\mathbf{u}\) points along the gradient; minimal, \(-|\nabla f|\), in the opposite sense; and zero in the perpendicular directions.

This explains the relation with level curves: along one of them \(f\) does not change, so the gradient is perpendicular to its tangent. For \(f=x^2+y^2\) at \((1,1)\), \(\nabla f=\langle2,2\rangle\) is normal to the circle \(x^2+y^2=2\) there. With that normal one writes the tangent line to a curve \(f(x,y)=c\): \(f_x(x-a)+f_y(y-b)=0\). In three variables the same argument gives the tangent plane to a level surface.

The gradient is defined the same way with three or more variables, as the vector of all partials: for \(f=xyz\), \(\nabla f=\langle yz,xz,xy\rangle\). Where it is the zero vector, the tangent plane is horizontal and the point is a candidate for a maximum, minimum or saddle.

A common mistake is to treat \(\nabla f\) as a number: it is a vector, and the direction of fastest increase requires normalizing it. Its immediate application is the directional derivative, which comes next.