Vector multiply that is NOT dot or cross?

In summary, the conversation discusses the use of del operator in studying vector identities and how to multiply vectors without the use of a dot or cross product. The speaker presents examples and clarifies the concept of scalar multiplication and the gradient of a scalar function. They also mention the use of geometric product and geometric derivative in Clifford algebras as a way to generalize div, grad, and curl operators.
  • #1
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Hi - just working through my text (studying by correspondence) on Del operator - so Curl, div etc. Came across some identities parts of which which have me confused. what does it mean when a vector is shown as multiplying something - but without dot or cross? For example F(∇.G) or ∇(F.G) or (G.∇)F ...

I get that something like (G.∇) expands to each component of G times each component of ∇ - which is a scalar; also ∇.G is a normal dot product. So I understand f.(∇.G) and ∇.(F.G) and (G.∇).F and but am confused when the 'dot' outside the bracket is missing - how do we multiply those?

Thanks
Alan
 
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  • #2
Are you seeing the gradient of a scalar function? That is represented by a del operator and a capital letter which you might think is a vector.
 
  • #3
These are all vectors, an example identity is
∇x(FXG) = F(∇.G) - G(∇.F) + (G.∇)F - (F.∇)G
 
  • #4
Just dot them together:

http://academics.smcvt.edu/jellis-monaghan/calc3/in%20class%20maple%20demos/graddivcurl1.pdf

Check out Example 6.6
 
  • #5
ognik said:
Hi - just working through my text (studying by correspondence) on Del operator - so Curl, div etc. Came across some identities parts of which which have me confused. what does it mean when a vector is shown as multiplying something - but without dot or cross? For example F(∇.G)
∇.G is a scalar function so F(∇.G) is "scalar multiplication"- each component of F multiplied by ∇.G

or ∇(F.G)
F.G is a scalar function so ∇(F.G) is the gradient of F.G

or (G.∇)F ...0
This is the same as G(∇.F)


I get that something like (G.∇) expands to each component of G times each component of ∇ - which is a scalar; also ∇.G is a normal dot product. So I understand f.(∇.G) and ∇.(F.G) and (G.∇).F and but am confused when the 'dot' outside the bracket is missing - how do we multiply those?

Thanks
Alan
 
  • Like
Likes ognik
  • #6
Nice explanation thanks hallsofivy, I could see dotting them was the only way to get anything done, but its nice to understand why.
 
  • #7
I'm not sure this is what you're looking for, but you might want to have a look at the definition of "geometric product" in Clifford algebras, plus the concept of "geometric derivative" proposed by D. Hestenes, which generalizes div,grad,curl operators.
 

1. What is vector multiplication?

Vector multiplication is a mathematical operation that involves multiplying two vectors together to produce a new vector. It is a fundamental operation in vector algebra and is used in a variety of scientific fields, including physics and engineering.

2. What are the different types of vector multiplication?

There are three types of vector multiplication: dot product, cross product, and scalar triple product. Dot product results in a scalar quantity, cross product results in a vector quantity, and scalar triple product results in a scalar quantity.

3. How is vector multiplication different from scalar multiplication?

Vector multiplication involves multiplying two vectors together, resulting in a new vector. Scalar multiplication involves multiplying a vector by a scalar, resulting in a scaled version of the original vector.

4. What are some applications of vector multiplication?

Vector multiplication is commonly used in physics and engineering to calculate forces, torque, and work. It can also be used in computer graphics to rotate, scale, and translate objects.

5. Can vector multiplication be performed on vectors of different dimensions?

No, vector multiplication can only be performed on vectors of the same dimension. This means that both vectors must have the same number of components for the operation to be valid.

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