12 - Gradient Vector

Gradient Vector

Let $f:=R^n\to \mathbb{R}$ be a scalar function whose partial derivatives all exist

Then the gradient vector is defined as

$$\nabla f=\left(\frac{\partial f}{\partial x_1},\frac{\partial f}{\partial x_2},\cdots ,\frac{\partial f}{\partial x_n}\right)$$

Note that $\nabla$ is “grad” or “del” or “nabla”

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Properties of $\nabla$ (nabla)

Let $f,g$ be differentiable functions of $x,y,z$ then

1. $\nabla (fg)=f\nabla g+g\nabla f$
2. $\nabla (f^n)=n{f}^{n-1}\nabla f$
3. $\nabla \left(\frac{f}{g}\right)=\frac{g\nabla f-f\nabla g}{g^2}$
4. $\nabla (f(g(\mathbf{x})))=f^{\prime }(g(\mathbf{x}))\nabla g(\mathbf{x})$