# What is Vector identities: Definition and 17 Discussions

The following are important identities involving derivatives and integrals in vector calculus.

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1. ### How to prove vector identities WITHOUT using levi civita ?

Mentor note: Thread moved from homework sections as being a better fit in the math technical section. Multiplying components of both sides are also off limits. I am trying to derive vector identities on introduction chapters in various EMT books. For example : (AXB).(CXD) = (A.C)(B.D) -...
2. ### Vector identities in quantum mechanics

The overall problem is to prove that [L^2,[L^2,\hat{r}]]=2\hbar^2 {L^2,r} I feel I am very close to solving this problem but I need a quantum version of the vector identity ax(bxc). Because the relevant vectors are operators that don't commute, there is a problem. Does anybody know of a source...
3. ### How to prove that H_a and H_b are orthogonal?

1. Okay, so I am going to prove that \int H_a\cdot H_bdv=0 Hint: Use vector Identities H is the Magnetic Field and v is the volume. Homework Equations this this[/B] k_bH_b=\nabla \times E_b k_aH_a=\nabla \times E_a k is the wave vector and E is the electric field The Attempt at a...

Please excuse my copying this question in, minimising my input :-) Part (b): Let $a.b \times c =v$ Then $a'.b' \times c' = \left( \frac{b \times c}{v}\right) . \left( \frac{c \times a}{v} \times \frac{a \times b}{v}\right)$ $= v^{-1} \left(b \times c\right). \left[ \left(c \times... 5. ### Deriving the Vector Identity:$\nabla(\vec{A} \cdot \vec{B})\$

Homework Statement I'm trying to derive the vector identity: $$\nabla(\vec{A} \cdot \vec{B})$$Homework Equations $$\nabla(\vec{A} \cdot \vec{B})=(\vec{B} \cdot \nabla) \vec{A} + ( \vec{A} \cdot \nabla ) \vec{B} + \vec{B} \times (\nabla \times \vec{A})+ \vec{A} \times ( \nabla \times \vec{B})$$...
6. ### Prove the Following Vector Identities Part 1

Homework Statement \displaystyle \frac{d}{dt} (\vec u (t) \vec v (t))= \vec u (t)' \vec v(t)+\vec u(t) \vec v(t)' I know this is a product rule on the RHS but how does one prove it? Thanks
7. ### Coulomb's Law from vector identities

I can show that Coulomb's Law + superposition implies \nabla \cdot \mathcal {E} = \frac{\rho}{\epsilon_0} and \nabla \times \mathcal{E} = \mathbf{0}. I want to go the other way and derive Coulomb's law and superposition from the vector identities. I know that Gauss' Law implies Coulomb's law if...
8. ### Need help proving vector identities

Homework Statement i have to prove that ∇x(FxG)=(G⋅∇)F-(F⋅∇)G+F(∇⋅G)-G(∇⋅F) where F and G are vector fields with F=F1,F2,F3 and G=G1,G2,G3 ∇=d/dx,d/dy/d/dz Homework Equations The Attempt at a Solution i have tried applying scalar multiplication and the cross product to...
9. ### Proving Vector Identities Using the Permutation Tensor and Kroenecker Delta

Homework Statement Prove using the Levi-Civita Tensor/Kroenecker Delta that: (AxB)x(CxD) = (A.BxD).C-(A.BxC).D Homework Equations εіјkεimn = δjmδkn – δjnδkm (where δij = +1 when i = j and 0 when i ≠ j) The Attempt at a Solution if E = (AxB) then Ei = εіјkAjBk, and if F =...
10. ### Proving vector identities

Homework Statement Prove the following vector identity: \nablax(AxB) = (B.\nabla)A - (A.\nabla)B + A(\nabla.B) - B(\nabla.A) Where A and B are vector fields. Homework Equations Curl, divergence, gradient The Attempt at a Solution I think I know how to do this: I have to...
11. ### Proving vector identities using Cartesian tensor notation

Homework Statement 1. Establish the vector identity \nabla . (B x A) = (\nabla x A).B - A.(\nabla x B) 2. Calculate the partial derivative with respect to x_{k} of the quadratic form A_{rs}x_{r}x_{s} with the A_{rs} all constant, i.e. calculate A_{rs}x_{r}x_{s,k} Homework Equations The...
12. ### Proving vector identities with index notation (help with the del operator)

Homework Statement Prove the vector identity: \left(a\times\nabla\right)\bullet\left(u \times v\right)=\left(a \bullet u \right)\left(\nabla \bullet v \right)+\left(v \bullet \nabla \right)\left(a \bullet u \right)-\left(a \bullet v \right)\left(\nabla \bullet u \right)-\left(u...
13. ### Vector identities in index notation

Homework Statement Prove using index notation that, the x denoting a cross-product. (del x f del g)=del f x del g Homework Equations The Attempt at a Solution dif etc. denote partial derivatives. RHS=eijkdjfdkg LHS-I'm not even quite sure how to write it in index...
14. ### Vector Identities: Calculate \nabla \cdot (f \nabla \times (f F))

Vector Identities ?? Having heaps of trouble with v.identities any help possible would be greatly appreciated. Let F = (z,y,-x) and f = |F| <--- (magnitude F) Use vector identities to calculate; \nabla \cdot (f \nabla \times (f F))
15. ### How Can I Prove Vector Identities Using Algebraic Manipulation?

Homework Statement Question One: Prove that |u x v|^2 = (u . u)(v . v)-(u . v)^2 where u and v are vectors. Question Two: Given that u = sv + tw, prove algebraically that u . v x w = 0 where u, v and w are vectors and s and t are integers. Homework Equations I don't know :( The...
16. ### Obtaining vector identities

Homework Statement The vectors F and G are arbitrary functions of position. Starting w/ the relations F x (∇ x G) and G x (∇ x F), obtain the identity ∇(F . G) = (F . ∇)G + (G . ∇)F + F x (∇ x G) + G x (∇ x F) Homework Equations The Attempt at a Solution I started off...
17. ### Vector Identities: Calculate & Surface Integrals

Homework Statement 1. Calculate: \nabla \times (\frac{\vec{p} \times \vec{r}}{r^{3}}) in cartesian and spherical coordinates, where \vec{p} is a constant vector. 2. Calculate surface integrals: \int \vec{r} (\vec{a} \cdot \vec{n}) dS \int \vec{n} (\vec{a} \cdot \vec{r}) dS where...