# Vector triple product causing a contradiction in this proof

• kostoglotov
Therefore, expanding the original statement using the vector triple product does not result in an equivalent statement.
kostoglotov

## Homework Statement

Prove the following identity

$$\nabla (\vec{F}\cdot \vec{G}) = (\vec{F}\cdot \nabla)\vec{G} + (\vec{G}\cdot \nabla)\vec{F} + \vec{F} \times (\nabla \times \vec{G}) + \vec{G}\times (\nabla \times \vec{F})$$

## Homework Equations

vector triple product

$$\vec{a} \times (\vec{b} \times \vec{c}) = \vec{b}(\vec{a}\cdot \vec{c}) - \vec{c}(\vec{a}\cdot \vec{b})$$

## The Attempt at a Solution

The first thing I wanted to do was investigate what expanding according to the vector triple product would do to the original statement I am trying to prove. This happens:

$$\nabla (\vec{F}\cdot \vec{G}) = (\vec{F}\cdot \nabla)\vec{G} + (\vec{G}\cdot \nabla)\vec{F} + \nabla (\vec{F}\cdot \vec{G}) - (\vec{F}\cdot \nabla)\vec{G} + \nabla (\vec{F}\cdot \vec{G}) - (\vec{G}\cdot \nabla)\vec{F} = 2\nabla (\vec{F}\cdot \vec{G})$$

What's happening here? Is it not valid to use the vector differential operator in an expansion of the vector triple product? Why not?

It may be because the vector operator does not commute in the dot product like ordinary vectors.

kostoglotov

## 1. How does the vector triple product cause a contradiction in this proof?

The vector triple product is a mathematical operation that involves three vectors, and is used to calculate the volume of a parallelepiped (a three-dimensional figure with six parallelogram faces). In this proof, the contradiction arises when the vector triple product of three vectors does not equal zero, which is necessary for the parallelepiped to have zero volume. This means that the three vectors are not linearly independent, and the proof cannot hold.

## 2. Can you explain the concept of linear independence in relation to the vector triple product?

Linear independence refers to the relationship between vectors in a given set. Two vectors are linearly independent if neither of them can be expressed as a linear combination of the other. In the context of the vector triple product, if the result is not equal to zero, it means that the three vectors involved are not linearly independent and cannot form a parallelepiped, leading to a contradiction in the proof.

## 3. Is the vector triple product always involved in proofs that lead to contradictions?

No, the vector triple product is not always involved in proofs that lead to contradictions. It is a mathematical operation that is used in certain situations, and its presence in a proof does not necessarily mean that there will be a contradiction. However, in cases where the vector triple product is used to calculate the volume of a parallelepiped, a contradiction can occur if the result is not equal to zero.

## 4. How does the contradiction in this proof impact the validity of the proof?

The contradiction in this proof means that the proof is not valid. A contradiction occurs when a statement or assumption leads to a logical inconsistency or impossibility, which means that the proof is flawed and cannot be considered true. In this case, the use of the vector triple product leads to a contradiction, rendering the proof invalid.

## 5. Are there any other mathematical operations that can cause contradictions in proofs?

Yes, there are other mathematical operations that can cause contradictions in proofs. The most common ones include division by zero, taking the square root of a negative number, and assuming that a number is both positive and negative at the same time. These contradictions can arise in various mathematical proofs and must be carefully avoided to ensure the validity of the proof.

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