Gradient of A*B: Adding and Subtracting Terms

In summary, the gradient of A*B is the rate of change of the product of two variables, A and B. It is calculated by taking the partial derivatives of A and B with respect to each variable and multiplying them together. The gradient can be positive or negative, indicating an increasing or decreasing rate of change. Adding or subtracting terms in the expression A*B will change the overall equation and therefore, the gradient. It is important in science because it helps us understand the relationship between variables and is commonly used in fields such as physics, engineering, and economics.
  • #1
xaos
179
4
grad(A*B)=(A*grad)B + (B*grad)A + A curl B + B curl A

i'm not sure how to read the RHS to begin to work out the index definition.i'm thinking if add and subtract terms this will work out. i think i can see the first two terms, but the last two maybe "A cross nabla" is what they mean acting on B componentwise, otherwise it doesn't mean anything to me.
 
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  • #2
This is the correct identity:

∇(A⋅B) = A⋅∇B + B⋅∇A + A x (∇ x B) + B x (∇ x A)
 

Related to Gradient of A*B: Adding and Subtracting Terms

1. What is the gradient of A*B?

The gradient of A*B refers to the rate of change of the product of two variables, A and B. It is a vector that represents the direction and magnitude of the steepest slope at a specific point on the graph of A*B.

2. How is the gradient of A*B calculated?

The gradient of A*B is calculated by taking the partial derivatives of A and B with respect to each variable and then multiplying them together. For example, if A*B = xy, the gradient would be (y, x).

3. Can the gradient of A*B be positive or negative?

Yes, the gradient of A*B can be positive or negative. A positive gradient indicates an increasing rate of change, while a negative gradient indicates a decreasing rate of change.

4. How does adding or subtracting terms affect the gradient of A*B?

Adding or subtracting terms in the expression A*B will change the overall equation, and therefore, the gradient will also change. The new gradient will be a combination of the gradients of the individual terms.

5. Why is the gradient of A*B important in science?

The gradient of A*B is important in science because it helps us understand the relationship between two variables and how they change in relation to each other. It is commonly used in fields such as physics, engineering, and economics to analyze and model real-world phenomena.

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