Multivariable Version of FToC?

In summary, the Multivariable Version of FToC is a mathematical theorem that relates integration and differentiation in multiple variables. It allows for the evaluation of integrals in multiple variables and provides a connection between these two fundamental concepts in calculus. This version differs from the single variable version by applying to functions of multiple variables and using multiple integral signs. It has various applications in fields such as physics, engineering, and economics. Common mistakes when applying this version include forgetting to include limits of integration, not recognizing the continuity of the function, and incorrectly setting up the integral.
  • #1
CSteiner
31
0
So for a function of a single variable

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How can this be extended to the integration of the total differential of a multivariable function over a region (specifically one of two variables)?
That is, how do you integrate

%20f%20%7D%7B%5Cpartial%20x%7Ddx%20+%20%5Cfrac%7B%5Cpartial%20f%20%7D%7B%5Cpartial%20y%7Ddy.gif


Say over the circular region

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,
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  • #2
The generalisation to this to an arbitrary number of dimensions is Stokes' theorem.
 
  • #3
Ah, thanks. I guess I'm going to have to get off my ass and finish that MIT OCW Multivariable Calculus course I've been studying. I'm about halfway through, so I've seen double integrals and differentials, but not stoke's theorem.
 

1. What is the Multivariable Version of FToC?

The Multivariable Version of FToC (Fundamental Theorem of Calculus) is a mathematical theorem that relates the concept of integration to the concept of differentiation in multiple variables. It states that if a function f(x,y) is continuous on a region R and has a continuous partial derivative with respect to x and y, then the integral of f(x,y) over R can be evaluated by finding an antiderivative of the partial derivative of f(x,y) with respect to x and y.

2. What is the significance of the Multivariable Version of FToC?

The Multivariable Version of FToC is significant because it allows for the evaluation of integrals in multiple variables, which is essential in many fields such as physics, engineering, and economics. It also provides a connection between integration and differentiation, which are two fundamental concepts in calculus.

3. How is the Multivariable Version of FToC different from the single variable version?

The single variable version of FToC only applies to functions of one variable, while the multivariable version applies to functions of multiple variables. Additionally, the single variable version uses only one integral sign while the multivariable version uses multiple integral signs to represent the integration over different variables.

4. What are some applications of the Multivariable Version of FToC?

The Multivariable Version of FToC has many applications in various fields, such as calculating volumes and areas of irregular shapes, finding the center of mass of an object, and solving optimization problems in physics and economics. It is also used in vector calculus to calculate line integrals and surface integrals.

5. What are some common mistakes made when applying the Multivariable Version of FToC?

One common mistake is forgetting to include the appropriate limits of integration for each variable. Another mistake is not recognizing when the function being integrated is not continuous or does not have a continuous partial derivative. It is also important to properly set up the integral by identifying the correct function to integrate and the appropriate region of integration.

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