# Technical question in multi-variable differentiation

• jjou
In summary, if a function f(x+iy) = u(x,y)+iv(x,y) is holomorphic on an open set and |f| is constant, then f is also constant. This is shown by using the fact that the squared magnitude of f, |f|^2 = u^2+v^2, is constant and the derivatives of this function with respect to x and y are both equal to zero. However, the statement (u^2)_x+(v^2)_y = 0 may not necessarily be true, as it depends on the specific functions u and v.
jjou
Let $$f(x+iy)=u(x,y)+iv(x,y)$$. Suppose we know $$|f|^2=u^2+v^2$$ is a constant function. Then we are allowed to say that $$(u^2+v^2)_x=(u^2+v^2)_y=0$$. But are we allowed to differentiate u by x and v by y? IE, are we allowed to make the following statement:
$$(u^2)_x+(v^2)_y=0$$

I'm guessing 'no', but I'm not too sure why. Intuitively, I would guess that you could change u and v in such a way that those changes balance each other out? (Very unclear way to say it...)

no youre not allowed.

add the functions together and differentiate it with respect to whatever subscipt it is

Not you're not allowed. Consider for instance the functions u²=2x+3y and v²=-2x-3y. Then $$(u^2+v^2)_x=(u^2+v^2)_y=0$$, but $$(u^2)_x+(v^2)_y=-1$$

jjou said:
Let $$f(x+iy)=u(x,y)+iv(x,y)$$. Suppose we know $$|f|^2=u^2+v^2$$ is a constant function. Then we are allowed to say that $$(u^2+v^2)_x=(u^2+v^2)_y=0$$. But are we allowed to differentiate u by x and v by y? IE, are we allowed to make the following statement:
$$(u^2)_x+(v^2)_y=0$$

I'm guessing 'no', but I'm not too sure why. Intuitively, I would guess that you could change u and v in such a way that those changes balance each other out? (Very unclear way to say it...)
Perhaps you could but that has nothing to do with the derivative. My question is why on Earth would you even consider that $u^2_x+ v^2_y= 0$?

My friend and I used that in a complex analysis proof that, for a function $$f(x+iy) = u(x,y)+iv(x,y)$$ that is holomorphic on an open set, if |f| is constant then f is constant.

If |f| is constant, then $$|f|^2 = u^2+v^2$$ is constant. Then the derivatives $$(u^2+v^2)_x$$ and $$(u^2+v^2)_y = 0$$.

We then also used the fact that $$(u^2)_x+(v^2)_y = 0$$ and manipulated the three equations using the Cauchy-Riemann equations to show that all partials were equal to zero ($$u_x = u_y = v_x = v_y = 0$$).

Will have to rethink the proof...

## What is multi-variable differentiation?

Multi-variable differentiation is the process of finding the rate of change of a function with respect to multiple independent variables. It involves calculating partial derivatives for each variable and then combining them to find the overall rate of change.

## How is multi-variable differentiation different from single-variable differentiation?

In single-variable differentiation, we only consider one independent variable. However, in multi-variable differentiation, we consider multiple independent variables and their rates of change simultaneously.

## What is the chain rule in multi-variable differentiation?

The chain rule in multi-variable differentiation is a method used to find the derivative of a composite function with multiple variables. It states that the derivative of a composite function is equal to the product of the derivatives of its individual functions.

## What is the purpose of multi-variable differentiation in science?

Multi-variable differentiation is used in many scientific fields such as physics, engineering, and economics to analyze and model complex systems. It allows us to understand how a system changes with respect to multiple variables and make predictions and optimizations based on these changes.

## What are some real-life applications of multi-variable differentiation?

Multi-variable differentiation is used in many real-world applications such as calculating the optimal production level in economics, determining the trajectory of a projectile in physics, and optimizing the shape of an airplane wing in engineering. It is also used in machine learning and data analysis to model and analyze complex datasets with multiple variables.

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