# Question about Cross Product and Derivative

• I
• bwest121
In summary, the cross product is a mathematical operation that takes two vectors as input and produces a third vector that is perpendicular to both of the input vectors. It is calculated using a specific formula and has a geometric interpretation as the area of a parallelogram. The cross product is different from the dot product in its properties and results, and it is commonly used in physics and engineering for various calculations and applications.
bwest121
Hi everyone,

Given a vector-valued function ##\vec{A}##, how do I show that:

$$\vec{\nabla} \times \left(\frac{\partial \vec{A}}{\partial x}\right) = \frac{\partial}{\partial x}(\vec{\nabla} \times \vec{A})$$

In other words, are the cross product and derivative commutative w/ each other? I have an intuition that this is true, but I would like to know a good way to show this.

Thank you very much.

If in doubt, write it out component by component and check.

Assuming the components of $\mathbf{A}$ are sufficiently smooth, partial derivative operators will commute.

## 1. What is the cross product?

The cross product, also known as the vector product, is a mathematical operation that takes two vectors as input and produces a third vector that is perpendicular to both of the input vectors.

## 2. How is the cross product calculated?

The cross product of two vectors, A and B, is calculated using the formula A x B = (AyBz - AzBy)i + (AzBx - AxBz)j + (AxBy - AyBx)k, where i, j, and k are unit vectors in the x, y, and z directions, respectively.

## 3. What is the geometric interpretation of the cross product?

The magnitude of the cross product of two vectors is equal to the area of the parallelogram formed by the two vectors, and the direction of the cross product is perpendicular to the plane of the parallelogram.

## 4. What is the difference between the cross product and the dot product?

The cross product and the dot product are both ways to combine two vectors, but they have different properties and produce different results. The dot product is a scalar quantity and represents the projection of one vector onto another, while the cross product is a vector quantity and represents the area and direction of the parallelogram formed by the two vectors.

## 5. How is the cross product used in physics and engineering?

The cross product is used in physics and engineering to calculate torque, angular momentum, and magnetic fields. It is also used in applications such as computer graphics and robotics to determine the orientation and motion of objects in three-dimensional space.

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