What is the difference between the Laplacian and second order derivative?

Question

Is the Laplacian of a function the same as second order derivative of the function in 1-D? What about in 2-D?

Answer 1

The Laplacian is the divergence of a gradient

$$\Delta f:= \nabla \cdot \nabla f.$$

If the function f is depending on the variables $x_1,x_2,\ldots,x_n$ then we can see that

$$\Delta f =\sum_{i=1}^{n}\dfrac{\partial^2 f}{\partial x_i\partial x_i}.$$

If $f=f(x_1)$ then the Laplacian is the second derivative. For $f=f(x_1,x_2)$ the Laplacian is given as:

$$\Delta f =\dfrac{\partial^2 f}{\partial x_1 \partial x_1}+\dfrac{\partial^2 f}{\partial x_2 \partial x_2}.$$