Toronto Math Forum
MAT3342018F => MAT334Lectures & Home Assignments => Topic started by: Yuhan Shao on October 16, 2018, 10:01:01 PM

The attachments are two different partial derivatives of f = u + iv respective to x and y
Is there a typo or I am missing something?
The one in the 2.1 seems correct.
Thanks!

It seems like the equation in 2.1 is
$$\frac{df(z)}{dz}$$
while the equation in 1.6 is
$$\frac{\partial f(p,q)}{\partial x} \text{ and } \frac{\partial f(p,q)}{\partial y}$$
Basically, the 2.1 equation talks about the derivative while 1.6 equations talk about the two separate partial derivatives.

There is no typo. Also, we can view $u$ as $\mathfrak{Re}f$ $v$ as $\mathfrak{Im}f$. The equation in 2.1 is the total derivative, whereas the equation in 1.6 are the two partial derivatives, one in respect to the real values $\frac{\delta u}{\delta x} + i\frac{\delta v}{\delta x}$ and the other in respect to the imaginary values $\frac{\delta u}{i\delta y} + \frac{+\delta v}{\delta y}$ (note that in the limit definition, you have to watch out for the $i$ on the denominator $k$).
The total derivative can be expressed as $\frac{df}{dz} = \frac{\delta u}{\delta x}\frac{dx}{dz} + i\frac{\delta v}{\delta x}\frac{dx}{dz}  i\frac{\delta u}{\delta y}\frac{dy}{dz} + \frac{\delta v}{\delta y}\frac{dy}{dz}$