Understanding Complex Operators

Question

I am not a follower of Schaum's Outline of Complex Variables but for seeking my professor lecture I read the third chapter called Complex Differentiation and the Cauchy-Riemann Equations. And I introduced some of the complex operators which isn't explained well manner in that chapter. Here are those:

If $A(x,y)=P(x,y)+iQ(x,y)$ is a complex continuously differentiable function of $x$ and $y$ (vector? isn't those come from $\mathbb R$) then $$\text{grad }A=\nabla A=\frac{\partial P}{\partial x}-\frac{\partial Q}{\partial y}+i\left(\frac{\partial P}{\partial y}+\frac{\partial Q}{\partial x}\right)$$

The $\nabla$ del symbol can be interpreted as a vector of partial derivative operators, and its three possible meanings—gradient, divergence, and curl—can be formally viewed as the product with a scalar, a dot product, and a cross product, respectively, of the del "operator" with the field. But I can't relate the above operation with any of those mentioned category (linked statement from Wikipedia). Rather it seems complex multiplication of two number.

The divergence of $A$ is, $$\text{div }A=\nabla\cdot A=\text{Re}\{\bar\nabla A\}=\text{Re}\left\{\left(\frac{\partial}{\partial x}-i\frac{\partial}{\partial y}\right)(P+iQ)\right \}$$

The curl of $A$ is, $$|\text{curl }A|=|\nabla \times A|=|\text{Im}\{\bar \nabla A\}|=\left|\text{Im}\left\{\left(\frac{\partial}{\partial x}-i\frac{\partial}{\partial y} \right)(P+iQ) \right\} \right|=\left| \left(\frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y} \right)\right|$$

There is no reference to how those formulas come from. I really want to know their derivation. Any reference or solution would be great helpful for me.

Answer 1

In this context the nabla operator $\nabla$ is a vector differential operator defined for complex functions. Intuitively, if we have $$ \nabla \equiv \left( \frac{\partial}{\partial x}, \frac{\partial}{\partial y} \right) $$ in $ \mathbb{R}^2 $. Then the duple, viewed in the complex number sense, can be rewritten as $$ \nabla \equiv \frac{\partial}{\partial x}+ i \frac{\partial}{\partial y}. $$ By this virtue, we may also define the following: $$ \bar\nabla \equiv \frac{\partial}{\partial x} - i \frac{\partial}{\partial y}. $$ A few remarks can be made concerning the definitions above:

1. They are differential operators, which implies that they don't carry any meaning unless they are applied to a complex function.
2. The definitions of divergence and curl coincides with that for functions in $ \mathbb{R}^2 $ (you may check by direct computations).
3. The definition of gradient follows from multiplying $ \nabla $ and the function itself. However, it no longer carries the definition of gradient in $ \mathbb{R}^2 $.