# Help with Beginner Index Notation

• fttteotd
In summary, the conversation discusses learning some basic index notation and a question about showing that curl(fF) = fcurl(F) + (∇f) x F. The solution attempted uses the chain rule to expand the expression.
fttteotd
Okay, so I'm learning some basic index notation, and I have a few questions...

## Homework Statement

f= scalar field
F = vector field

so, we are supposed to show that curl(fF) = fcurl(F) + ($$\nabla$$f) x F

## The Attempt at a Solution

curl(fF) = [$$\nabla$$ x (fF))]$$_{k}$$ = $$\in$$$$_{ijk}$$(f$$\partial$$$$_{i}$$F$$_{j}$$ + F$$_{j}$$$$\partial$$$$_{i}$$f)

any help??

(all the superscripts are supposed to be subscripts, i dunno)

Welcome to PF!

Hi fttteotd! Welcome to PF!

(have a curly d: ∂ and a nabla: ∇ and an epsilon: ε and use itex rather than tex in the middle of a line )

[∇ x (fF)]i = εijkj(fFk) …

and now use the chain rule!

Sure, I'd be happy to help with your beginner index notation questions. It looks like you're on the right track with your attempt at a solution. Let me break down the steps for you:

1. First, let's define the curl of a vector field F as the vector field given by the cross product of the gradient operator (\nabla) and F. In index notation, this can be written as [\nabla x F]_{k} = \in_{ijk}\partial_{i}F_{j}.

2. Next, we can use the product rule for differentiation to expand the expression for curl(fF). This gives us [\nabla x (fF))]_{k} = f[\nabla x F]_{k} + F_{j}[\nabla x f]_{k}.

3. Now, we can substitute our definition of the curl of a vector field from step 1 into the first term of the expanded expression. This gives us [\nabla x (fF))]_{k} = f(\in_{ijk}\partial_{i}F_{j}) + F_{j}[\nabla x f]_{k}.

4. We can then use the product rule again to expand the second term, [\nabla x f]_{k}, which gives us [\nabla x (fF))]_{k} = f(\in_{ijk}\partial_{i}F_{j}) + F_{j}(\in_{ijk}\partial_{i}f).

5. Now, we can rearrange the terms in the second term to get [\nabla x (fF))]_{k} = f(\in_{ijk}\partial_{i}F_{j}) + (\in_{ijk}F_{j}\partial_{i}f).

6. Finally, we can use the definition of the cross product to rewrite the first term, \in_{ijk}\partial_{i}F_{j}, as the curl of the vector field F. This gives us [\nabla x (fF))]_{k} = f(\in_{ijk}\partial_{i}F_{j}) + (\nabla x F)_{k}\cdot f.

7. And there we have it, we have shown that curl(fF) = fcurl(F) + (\nabla x F) x f, as required.

## 1. What is index notation?

Index notation is a mathematical notation used to represent repeated multiplication in a concise and organized way. It uses indices, or small numbers written above and to the right of the base number, to indicate the number of times the base number is multiplied by itself.

## 2. How do I read or pronounce indices?

Each index is read as "to the power of." For example, 23 is read as "two to the power of three," or simply "two cubed."

## 3. Can I use index notation with any type of numbers?

Yes, index notation can be used with any type of numbers, including whole numbers, fractions, decimals, and even negative numbers.

## 4. What are the basic rules of index notation?

The basic rules of index notation include the product rule, which states that when multiplying two numbers with the same base, you add the indices together; the power rule, which states that when raising a power to another power, you multiply the indices; and the quotient rule, which states that when dividing two numbers with the same base, you subtract the indices.

## 5. How can I simplify expressions with index notation?

To simplify expressions with index notation, you can use the rules mentioned above and follow the order of operations. Start by simplifying any operations inside parentheses, then exponents, then multiplication and division from left to right, and finally addition and subtraction from left to right.

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