Thread: Gradient

1. Gradient

The gradient is just the jacobian matrix correct

$f(x,y) = \log \sqrt{x^2 + y^2}$

$\nabla f(x,y) = \left[\begin{array}{cc} \frac{1}{\ln 10}\frac{x}{x^2+y^2}&\frac{1}{\ln 10} \frac{y}{x^2 +y^2} \end{array}\right]$ ??

also, I went to a couple differential geometry talks before and they were referring to nabla notation, were they referring to the gradient?

2. Re: Gradient

The gradient of a function of two variables is the vector $\frac{\partial f}{\partial x}\vec{i}+ \frac{\partial f}{\partial y}\vec{j}$. I'm not sure I would call that the "Jacobian matrix". The "nabla" is the $\nabla$ operator which can be thought of a kind of "symbolic vector": $\frac{\partial}{\partial x}\vec{i}+ \frac{\partial}{\partial y}\vec{j}$ or, in three dimensions $\frac{\partial}{\partial x}\vec{i}+ \frac{\partial}{\partial y}\vec{j}+ \frac{\partial}{\partial z}$.

The gradient can be thought of a product of that "vector" with a scalar function: $\nabla f= (\frac{\partial}{\partial x}\vec{i}+ \frac{\partial}{\partial y}\vec{j}+ \frac{\partial}{\partial z})f$ $= \frac{\partial f}{\partial x}\vec{i}+ \frac{\partial f}{\partial y}\vec{j}+ \frac{\partial f}{\partial z}\vec{k}$.

But you can also take the dot product of the "vector" operator and a vector valued function: $\nabla\cdot \vec{F}= (\frac{\partial}{\partial x}\vec{i}+ \frac{\partial}{\partial y}\vec{j}+ \frac{\partial}{\partial z}\vec{k})\cdot (f(x,y,z)\vec{i}+ g(x,y,z)\vec{j}+ h(x,y,z)\vec{k})$ $= \frac{\partial f}{\partial x}+ \frac{\partial g}{\partial y}+ \frac{\partial h}{\partial z}$. That is the "divergence of the vector valued function" or $div(\vec{F})$.

Or you can take the cross product of the "vector" operator and a vector valued function: $\nabla\times \vec{F}= ((\frac{\partial}{\partial x}\vec{i}+ \frac{\partial}{\partial y}\vec{j}+ \frac{\partial}{\partial z}\vec{k})\times (f(x,y,z)\vec{i}+ g(x,y,z)\vec{j}+ h(x,y,z)\vec{k})= \left(\frac{\partial h}{\partial y}- \frac{\partial g}{\partial z}\right)\vec{i}+ \left(\frac{\partial f}{\partial z}- \frac{\partial h}{\partial x}\right)\vec{j}+ \left(\frac{\partial g}{\partial x}- \frac{\partial f}{\partial y}\right)\vec{k}$.
That is the "curl of the vector valued function" or $curl(F)$.

3. Re: Gradient

okay cool, I can see the similarity between derivative(jacobian) matrix and gradient.

thanks