# Total differential integration help

• TromboneNerd
In summary, you were asking how to integrate a total derivative, but when you try to integrate it, the right side is equal to z times the number of dimensions you're dealing with. This is because the total derivative is a function of both x and y, so when you anti-differentiate it, you are anti-differentiating it with respect to x and y.
TromboneNerd
Ok, so I'm having a little trouble with total differentiation. I know the total derivative is:

$$dz=\frac{}{}\partial z/\partial x dx+\frac{}{}\partial z/\partial y dy$$

but when i try to integrate it, the right side of the equation is equal to z times the number of dimensions you're dealing with.

$$\int dz=\int\frac{}{}\partial z/\partial x dx+\int\frac{}{}\partial z/\partial y dy$$

$$z=z+z$$

Is there something unique in total derivatives that I am missing that doesn't allow me to integrate like this? or am i making a horrible assumption with integrating partial derivatives in that they are the same as regular derivatives when integrating in respect to the same thing? My high school calculus teacher can't help me because he hasn't done this kind of math since college, which was quite some time ago. This isn't a homework problem (my actual homework is quite boring), just something i ran into on accident while studying ahead in my book that has me confused.

And i just figured out what i did wrong in the latex, so don't even mention it. this is what i meant, though i really doesn't change anything, it just looks better.

$$\int dz=\int\frac{\partial z}{\partial x}dx+\int\frac{\partial z}{\partial y}dy$$

Welcome to PF!

Hi TromboneNerd! Welcome to PF!

(have a curly d: ∂ and an integral: ∫ )

TromboneNerd said:
… when i try to integrate it, the right side of the equation is equal to z times the number of dimensions you're dealing with.

If you write the limits to the integrals, you'll see why it doesn't work.

z is a function of both x and y, so in the first integral, what limits are you integrating between?

and what are your limits for the second and third integrals, in x and y respectively?

An integral sign has no meaning on its own -- just affixing one to both sides should be somewhat suspect.

Normally, when you anti-differentiate, you are anti-differentiating a function respect to a variable, and you stick $\int \,\,\,\, dz$ around the function to indicate what is being anti-differentiated and with respect to what variable*.

But you're not anti-differentiating a function with respect to a variable; you said you want to anti-differentiate a form. Are you sure that's really what you want? I don't think I've ever seen anti-derivatives of forms presented as if that was an integral-like operation.

Anyways, only exact forms have anti-derivatives... dz is exact, but $\partial z/\partial x \, dx$ is rarely so.

*: There are some awkward issues involved when you have more than one independent variable.

I should have put a constant of integration somewhere because I don't have any limits set up. is there a reason why indefinite integration won't work? It can't be in the constant, because that cancels if i do put limits on it. but about the z being a function of x and y, i think that is what i was overlooking. is this what it "should" look like then?

$$z(x,y)=z(x)+z(y)$$

does this make any sense at all? again, I'm still relatively fresh in multivariable functions, so i am making a lot of assumptions.

I'm not sure what I said made any mathematical sense, but here's basically what I meant and it appears to work.

lets say z=2x+y, so

$$dz=\frac{\partial z}{\partial x}dx+\frac{\partial z}{\partial y}dy$$

$$dz=2dx+dy$$

$$\int dz=\int 2dx+\int dy$$

$$z=2x+y+C$$

So it was just my overlooking the fact that $$\int\frac{\partial z}{\partial x}dx$$ doesn't equal z, but just the x part of the function z.

Linear functions are extremely bad test cases when you're checking for odd behavior.

What kind of functions should I use then?

TromboneNerd said:
lets say z=2x+y
I think the linear function used here provides a good, simple counterexample to show what was wrong with the original question.

## 1. What is total differential integration?

Total differential integration is a mathematical concept used to calculate the total change in a function based on small changes in multiple variables. It is commonly used in physics and engineering to analyze complex systems.

## 2. How is total differential integration different from regular integration?

Regular integration calculates the area under a curve, while total differential integration calculates the total change in a function based on small changes in multiple variables. Total differential integration takes into account the interaction between these variables, whereas regular integration does not.

## 3. When is total differential integration used?

Total differential integration is used when analyzing complex systems with multiple variables, such as in physics and engineering. It is also commonly used in economics and finance to model changes in variables over time.

## 4. What is the formula for total differential integration?

The formula for total differential integration is Δf = (∂f/∂x)Δx + (∂f/∂y)Δy + (∂f/∂z)Δz + ..., where Δf is the total change in the function, ∂f/∂x, ∂f/∂y, and ∂f/∂z are the partial derivatives of the function with respect to each variable, and Δx, Δy, and Δz are the small changes in each variable.

## 5. How is total differential integration used in real-life applications?

Total differential integration has many real-life applications, such as in physics and engineering to analyze complex systems, in economics and finance to model changes in variables over time, and in machine learning and data analysis to understand the relationships between different variables. It is also used in optimization problems to find the maximum or minimum value of a function.

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