Yet another cross-product integral

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In summary, the conversation was about integrating a cross product and the possibility of it being true or not. The person also asked about the type of vector in the integral and clarified if they meant inner product or not.
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Johan_S
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I am trying to figure out how to do a more complex cross-product integral and get stuck, and since my books are 1000 km away I turn to here
I am trying to integrate a cross product and I wonder if the following is true. It does not feel like it is true but it would be very nice if it was since otherwise I have a problem with the signs...

This is my first time posting here, so I just pasted in the LaTeX code and hope that it is parsed...

##\int\overline{r} \times \frac{d\overline{p}}{dt} \; d\overline{\phi} = - \int\overline{r} \times \frac{d\overline{\phi}}{dt} \; d\overline{p}##
 
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What kind of vector ##\bar{\phi}## is ?

I observe two vectors in your integral
[tex]\int \mathbf{A} d\mathbf{B}[/tex].
Do you mean inner product
[tex]\int \mathbf{A} \cdot d\mathbf{B}[/tex] ?
 
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1. What is a cross-product integral?

A cross-product integral is a type of integral that involves taking the product of two vectors and integrating them over a certain region. It is commonly used in vector calculus and can be used to calculate the area, volume, or higher-dimensional equivalents of a region.

2. How is a cross-product integral different from a regular integral?

A cross-product integral involves vectors, while a regular integral involves scalar functions. This means that in a cross-product integral, the integrand is a vector function, while in a regular integral it is a scalar function. Additionally, a cross-product integral is used to calculate quantities related to a region, while a regular integral is used to calculate the area under a curve.

3. What are some applications of cross-product integrals?

Cross-product integrals have various applications in physics, engineering, and mathematics. They can be used to calculate the moment of inertia of an object, the work done by a force on a moving object, or the flux of a vector field through a surface. They are also used in the study of electromagnetism and fluid mechanics.

4. How do you solve a cross-product integral?

The process for solving a cross-product integral involves finding the limits of integration, setting up the integrand as a vector function, and then integrating each component of the vector separately. This can be done using various techniques such as substitution, integration by parts, or using trigonometric identities. It is important to carefully consider the geometry of the region and choose the appropriate coordinate system for the integral.

5. Are there any special properties of cross-product integrals?

Yes, there are a few special properties of cross-product integrals that can be useful in solving them. For example, the cross product of two vectors is perpendicular to both of the original vectors, which can help determine the orientation of the integral. Additionally, the cross product is anti-commutative, meaning that the order of the vectors in the cross product does not matter, which can simplify calculations.

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