If $f(\mathbf{r})=G(||\mathbf{r}||^2), \mathbf{r}\in \mathbb{R}^n $ and G is just some differentiable function then what would the gradient vector $\nabla f $ be?
I got it as $\nabla f = 2\mathbf{r} G'(||\mathbf{r}||^2) .$ is this correct?
Partial derivatives - simple
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multivariable-calculus
partial-derivative
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0Yes that is correct. – 2017-02-08
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0It might be worth pointing out here that since $\nabla f(\mathbf{r})$ is a scalar multiple of $\mathbf{r}$, $\nabla f$ is perpendicular to the sphere in $\mathbb{R}^n$ centered at the origin and containing $\mathbf{r}$. This satisfies our geometric understanding of the gradient - the level surfaces of $f$ (spheres) are perpendicular to $\nabla f$, and $\nabla f$ points in the direction of greatest increase. – 2017-02-09