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We just learned what the gradient of a function is. It means the largest change in a function. It is the directional derivative.

However I have also seen notation that lists the gradient squared of a function.

If I have $f(x,y)$, and take it gradient, I get $\partial f/\partial x$ i +$\partial f/\partial y$ j.

What does it mean and if someone could write out explicitly, what it means do take the gradient of this?

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1 Answer

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The operation $\nabla^2$ is the Laplace Operator $$\frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} + \frac{\partial^2}{\partial z^2}.$$ The reason for this notation is that it is common to write the usual gradient operator $\nabla$ out as $$\nabla = \frac{\partial}{\partial x}\mathbf{i} + \frac{\partial}{\partial y}\mathbf{j} + \frac{\partial}{\partial z}\mathbf{k},$$ so that $$\nabla^2 = \nabla \cdot \nabla = \left(\frac{\partial}{\partial x}\right)^2 + \left(\frac{\partial}{\partial y}\right)^2 + \left(\frac{\partial}{\partial z}\right)^2 = \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} + \frac{\partial^2}{\partial z^2}.$$ Anyway, if you apply $\nabla^2$ to a function, you get out $$\nabla^2 f = \frac{\partial^2f}{\partial x^2} + \frac{\partial^2f}{\partial y^2} + \frac{\partial^2f}{\partial z^2}.$$

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