Find Laplacian of the following field

Question : Find Laplacian of the following field 𝑓 = 𝑒^2𝑥 𝑠𝑖𝑛(2𝑦)

Solution :

Laplacian of a scalar field
\[f\ =\ e^{2x}\ sin\left(2y\right)\]is given by
\[\nabla^2f=\frac{\partial^2f}{\partial x^2}+\frac{\partial^2f}{\partial y^2}\]differentiating it partially with respect to x and y
\[\frac{\partial f}{\partial x}=2\ e^{2x}\ sin\left(2y\right)\ ,\ \ \frac{\partial f}{\partial y}=2\ e^{2x}\ cos\left(2y\right)\]
Again differentiating these partially with respect to x and y
\[\frac{\partial^2f}{\partial x^2}=4\ e^{2x}\ sin\left(2y\right)\ ,\frac{\partial^2f}{\partial y^2}=-4\ e^{2x}\ sin\left(2y\right)\]
Adding last two eqs
\[\nabla^2f=0 \]
We obtained Laplacian of the field 𝑓 = 𝑒^2𝑥 𝑠in(2y) is given as 0.