Can I use complex analysis to solve a vector calculus problem?

Question

This is a question from a non-mathematician, so excuse me if I use a more plain language.
Why can't I use complex analysis methods to solve a problem in vector calculus in 2 dimensions?
Say we are in 2 dimensions; we have unit vectors $\hat{i}$ and $\hat{j}$ at the $x$ and $y$ axis respectively. If we replace the $\hat{j}$ with $\hat{i}$ and we make the y axis the imaginary axis, then what is the difference between solving a problem with complex analysis methods and solving it with vector calculus methods?

Edit: I know that there should a difference but I do not know what it is.

Answer 1

There's a direct correspondence between vector fields on the 2d plane and complex functions. Hence, problems that can be described as integrals of the former (in particular, line integrals of vector fields) can be described as contour integrals in complex analysis instead.

As has been said, the notion of differentiation is different. In particular, those vector fields whose divergence and curl are zero correspond to complex functions that are complex differentiable. There is, however, a direct correspondence between $\nabla$ in vector calculus and $\partial/\partial \bar z$.

Answer 2

Some two-dimensional vector calculus problems can be solved using complex methods. It really depends on the problem. Some can easily be translated into the language of complex analysis, others not so easily.