# Vector, cross product, and integral

• foxjwill
In summary, the problem is to evaluate the integral of the cross product between vector fields \textbf{F} and \textbf{v} using \texttt{d}\textbf{v} as the differential and solving as three path integrals over a given path \textbf{r}.
foxjwill
[SOLVED] Vector, cross product, and integral

## Homework Statement

Evaluate:

$${\int \textbf{F} \times \texttt{d}\textbf{v}}.$$

$$\textbf{F}$$ and $$\textbf{v}$$ are both vector fields in $$\mathbb{R}^3$$

## Homework Equations

$$\texttt{d}\textbf{v} = (\nabla \otimes \textbf{v} ) \texttt{d}\textbf{r}$$

## The Attempt at a Solution

$$\begin{array}{ll} \textbf{F} \times \texttt{d}{\textbf{v}} &= \left( { \begin{array}{c} {F_2 \texttt{d}v_3 - F_3 \texttt{d}v_2 } \\ {F_1 \texttt{d}v_2 - F_2 \texttt{d}v_1 } \\ {F_1 \texttt{d}v_3 - F_3 \texttt{d}v_1 } \\ \end{array} \right ) \\ &= \left( \begin{array}{c} {F_2 \nabla v_3 \cdot \texttt{d}{\textbf{r}} - F_3 \nabla v_3 \cdot \texttt{d}{\textbf{r}}} \\ {F_1 \nabla v_2 \cdot \texttt{d}{\textbf{r}} - F_2 \nabla v_1 \cdot \texttt{d}{\textbf{r}}} \\ {F_1 \nabla v_3 \cdot \texttt{d}{\textbf{r}} - F_3 \nabla v_1 \cdot \texttt{d}{\textbf{r}}} \\ \end{array} \right) \\ \end{array}$$

This can then be solved as three path integrals over some path $$\textbf{r}$$. Is this correct?

Yes, that is correct.

## 1. What is a vector?

A vector is a mathematical object that has both magnitude (size or length) and direction. It is often represented graphically as an arrow, with the length of the arrow indicating the magnitude and the direction of the arrow indicating the direction.

## 2. What is the cross product of two vectors?

The cross product of two vectors is a vector that is perpendicular (or orthogonal) to both of the original vectors. It is calculated by taking the product of the magnitudes of the two vectors and the sine of the angle between them. The direction of the cross product can be determined using the right-hand rule.

## 3. How is the cross product useful in physics and engineering?

The cross product is useful in many applications, such as calculating torque, magnetic fields, and angular momentum. It can also be used to determine the orientation of a plane or the direction of rotation in a system.

## 4. What is an integral?

An integral is a mathematical concept that is used to calculate the area under a curve or the accumulated value of a changing quantity. It is represented by the symbol ∫ and can be thought of as the reverse of differentiation.

## 5. How is the integral used in real-world problems?

The integral is used in a wide range of real-world problems, such as finding the distance traveled by an object, calculating the work done by a force, and determining the probability of an event occurring. It is also used in many fields of science and engineering, including physics, chemistry, and economics.

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