- #1

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- Homework Statement
- $$A = (x^2, y^2, z^2)$$

$$B = (z, y, x)$$

Calculate $$grad(A \cdot B)$$

- Relevant Equations
- $$\nabla (A \cdot B) = (B \cdot \nabla)A + (A \cdot \nabla)B + B \times (\nabla \times A) + A \times (\nabla \times B)$$

Calculating dot product then doing gradient on it gets you:

$$(2xz + z^2, 3y^2, x^2 + 2xz)$$

which is the correct answer.

Using the formula, which you're required to do, gets a whole different answer.

Lets do each term individually.

##(B \cdot \nabla)A##

$$(B \cdot \nabla) = 1$$

## (A \cdot \nabla)B##

$$(A \cdot \nabla) = 2(x + y + z)$$

For the cross product terms, ##(\nabla \times A)## and ##(\nabla \times B)## both gets you the zero vector, which cross with anything still just gives zero.

So you're left with

$$1A + 2(x + y + z)B = (x^2 + 2xz + 2yz +2z^2, 2xy + 3y^2 + 2yz, 2x^2 + 2xy + 2xz + z^2)$$

Which is nothing like the answer. I've recalculated every single piece of this equation 10 times and I can testify that the equation given in Relevant Equations is false.

$$(2xz + z^2, 3y^2, x^2 + 2xz)$$

which is the correct answer.

Using the formula, which you're required to do, gets a whole different answer.

Lets do each term individually.

##(B \cdot \nabla)A##

$$(B \cdot \nabla) = 1$$

## (A \cdot \nabla)B##

$$(A \cdot \nabla) = 2(x + y + z)$$

For the cross product terms, ##(\nabla \times A)## and ##(\nabla \times B)## both gets you the zero vector, which cross with anything still just gives zero.

So you're left with

$$1A + 2(x + y + z)B = (x^2 + 2xz + 2yz +2z^2, 2xy + 3y^2 + 2yz, 2x^2 + 2xy + 2xz + z^2)$$

Which is nothing like the answer. I've recalculated every single piece of this equation 10 times and I can testify that the equation given in Relevant Equations is false.