Gradient Vector

The gradient vector of a bivariate function is the rate of change of that function, in any 2-dimensional direction.

Definition §

Gradient Vector

$$\text{grad }f=\nabla f=(\frac{\partial f}{\partial x},\frac{\partial f}{\partial y})$$

Just like the derivate of a 1D (univariate) function requires an x value to return the derivate at that point, a gradient vector requires the coordinate $(a,b)$ to be useful. The gradient vector at the point $(a,b)$ is:

$$\nabla f(a,b)={\left(\frac{\partial f}{\partial x},\frac{\partial f}{\partial y}\right)|}_{(x,y)=(a,b)}$$

Applications §

The gradient vector always point towards the direction of steepest ascent (maximum gradient). The magnitude of the gradient vector also gives the steepness of the ascent.

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