Vector Analysis Identity derivation

In summary, the given identity is derived by setting F=<a,b,c> and taking the partial derivatives with respect to x, y, and z. The resulting components are then combined to show that a vector equals a vector plus a scalar. However, it is important to note that del(F) is a rank 2 tensor and its interpretation is ambiguous.
  • #1
EngageEngage
208
0

Homework Statement



derive the identity:
del((F)^2) = 2 F . del(F) + 2Fx (del x F)

the dot is a dot product

Homework Equations


The Attempt at a Solution


first i set F = <a,b,c>, making F^2 = a^2 + b^2 + c^2
I took the partial derivatives with respect to x, y, and z (to get the necessary parts for the gradient), which gives me:

d/dx(a^2 + b^2 + c^2) = 2a * da/dx + 2b*db/dx + 2c*db/dc
d/dy(...) = ... 2a * da/dy + 2b*db/dy + 2c*db/dy
d/dz(...) = ... 2a * da/dz + 2b*db/dz + 2c*db/dz

But, because its a gradient of a scalar, the three components above are those of a vector. After putting together the components I am not sure where to go. What actually confuses me is that in the derivation you are supposed to show that a vector equals a vector plus a scalar -- the ( 2F . del(F)) term is a scalar... . If someone could please help me out that would be greatly appreciated. Thank you
 
Physics news on Phys.org
  • #2
If F is a vector, then you can't interpret del(F) as a vector. It has to be a rank 2 tensor. It's ij component is del_i(F_j) or del_j(F_i). So the i component of F.del(F) would be F_j*del_i(F_j) (summed over j) or possibly F_j*del_j(F_i) (summed over k). It's kind of ambiguous. What does it mean??
 
  • #3
derivation

how can you do the derivation of d = Da/2b
 

1. What is vector analysis identity derivation?

Vector analysis identity derivation is a mathematical process used to derive relationships between different vector quantities, such as position, velocity, and acceleration. It is often used in physics and engineering to simplify complex vector equations and make them easier to solve.

2. What are the main vector analysis identities?

The main vector analysis identities include the dot product identity, the cross product identity, and the triple product identity. These identities are used to manipulate vector equations and express them in different forms.

3. How is vector analysis identity derivation used in real-world applications?

Vector analysis identity derivation is used in a variety of real-world applications, such as in the design of structures and machines, in the analysis of motion and forces in physics, and in the study of electromagnetic fields in engineering.

4. What are the steps involved in vector analysis identity derivation?

The first step in vector analysis identity derivation is to identify the vector quantities involved in the equation. Then, use the given identities and mathematical operations to manipulate the equation and simplify it. Finally, solve for the desired vector quantity.

5. Are there any tips for successfully deriving vector analysis identities?

Some tips for successfully deriving vector analysis identities include being familiar with the basic vector operations and identities, practicing with various equations and identities, and using geometric reasoning to visualize the relationships between vector quantities.

Similar threads

  • Calculus and Beyond Homework Help
Replies
4
Views
871
  • Calculus and Beyond Homework Help
Replies
1
Views
736
  • Calculus and Beyond Homework Help
Replies
3
Views
209
  • Calculus and Beyond Homework Help
Replies
8
Views
422
  • Calculus and Beyond Homework Help
Replies
6
Views
2K
  • Calculus and Beyond Homework Help
Replies
5
Views
2K
  • Calculus and Beyond Homework Help
Replies
4
Views
791
  • Calculus and Beyond Homework Help
Replies
4
Views
2K
  • Calculus and Beyond Homework Help
Replies
12
Views
965
  • Calculus and Beyond Homework Help
Replies
9
Views
728
Back
Top