Corollaries of the fundamental integral theorems

In summary, the corollaries of the fundamental integral theorems have significant applications in fluid dynamics and are particularly useful in formulating integral laws. However, it is important to use these formulas with caution and in Cartesian coordinates for accurate results.
  • #1
Pushoam
962
52
Can anyone please tell me significance of these corollaries of fundamental integral theorems?
I can prove these corollaries but I don't understand why do we need to learn it?
Do these corollaries have some physical significance?

(a)$$\iiint_V(\nabla T)d^3 x=\oint_S T d\vec a$$
here S is the surface bounding the volume V .
(b)$$\iiint_V(\nabla \times\vec v)d^3 x= -\oint_S\vec v\times d\vec a$$,where S is the surface bounding the volume V .

(c)$$\iint_S(\nabla T)\times d\vec a =-\oint_P T d\vec l$$ , where P is the boundary of the surface S.
.
 
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  • #2
It's sometimes useful in formulating integral laws in fluid dynamics. Be careful with these formulae. They are really safe only when used in Cartesian coordinates!
 
  • #3
o.k.
Thank you.
 

FAQ: Corollaries of the fundamental integral theorems

What are the fundamental integral theorems?

The fundamental integral theorems are a set of mathematical theorems that relate to the integration of functions. They include the fundamental theorem of calculus, the fundamental theorem of line integrals, and the fundamental theorem of surface integrals.

What are corollaries of the fundamental integral theorems?

Corollaries of the fundamental integral theorems are additional theorems that can be derived from the fundamental theorems. They often involve specific applications or extensions of the fundamental theorems.

What is the significance of the fundamental integral theorems?

The fundamental integral theorems are significant because they provide a powerful tool for evaluating integrals and solving various mathematical problems. They also have many practical applications in fields such as physics, engineering, and economics.

How can the fundamental integral theorems be applied?

The fundamental integral theorems can be applied in a variety of ways, such as calculating areas and volumes, finding the average value of a function, and solving differential equations. They can also be used to evaluate complex integrals and to prove other mathematical theorems.

Are there any limitations to the fundamental integral theorems?

While the fundamental integral theorems are powerful and versatile, they do have some limitations. For example, they may not be applicable to functions that are not continuous or to regions that are not simply connected. In these cases, other methods of integration may be necessary.

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