What is Vector identity: Definition and 38 Discussions

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

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  1. G

    I Deriving Curl of B from Biot-Savart Law & Vector Identity

    $$\nabla \times B(r)=\frac{\mu _0}{4\pi} \int \nabla \times J(r') \times \frac{ (r-r')}{|r-r|^3}dV'$$ using the vector identity: $$\nabla \times (A \times B) = (B \cdot \nabla)A - B(\nabla \cdot A) - (A \cdot \nabla )B + A(\nabla \cdot B)$$ ##A=J## and ##B=\frac{r-r'}{|r-r'|^3}## since...
  2. C

    A Angular Moment Operator Vector Identity Question

    In my EM class, this vector identity for the angular momentum operator (without the ##i##) was stated without proof. Is there anywhere I can look to to actually find a good example/proof on how this works? This is in spherical coordinates, and I can't seem to find this vector identity anywhere...
  3. Faisal Moshiur

    I Proof of some identities regarding spin angular momentum.

    If we define Si=(1/2)× (reduced Planck's const)×sigma Then what will be (sigma dot vect{A})multiplied by (Sigma dot vect{B}) Here (sigma)i is Pauli matrix. Next one is, what will we get from simplifying <Alpha|vect{S}|Alpha> where vect{S} is spin vector & |Apha>is equal to " exp[{i×(vect{S} dot...
  4. Xsnac

    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 $$ \nabla(\vec{A} \cdot \vec{B})=(\vec{B} \cdot \nabla) \vec{A} + ( \vec{A} \cdot \nabla ) \vec{B} + \vec{B} \times (\nabla \times \vec{A})+ \vec{A} \times ( \nabla \times \vec{B})$$...
  5. D

    Proving Vector Identity Using Tensors: Urgent Help Needed

    Homework Statement Hello everyone, can anyone help me prove this using tensors? Given three arbitrary vectors not on the same line, A, B, C, any other vector D can be expressed in terms of these as: where [A, B, C] is the scalar triple product A · (B × C) Homework Equations I know that...
  6. E

    Vector identity proof in general curvilinear coordinates

    Homework Statement Need to prove that: ,b means partial differentation with respect to b, G is the metric tensor and Γ is Christoffel symbol. I think I could proceed with this quite well if I could understand the hint given, that I should lower the index j. Homework Equations am=Gmjaj...
  7. majormuss

    Vector identity question

    Does ∇⋅A = A ⋅∇? If not then, what does the latter actually equal?
  8. M

    Identity for Matrix*Vector differentiation w.r.t a vector

    I have J - matrix x and y - vector d [ J(x) y(x)] / dx I can multiply the matrix and vector together and then differentiate but I think for my application it would be better to find an identity like {d [ J(x) y(x)] / dx } = J(x) d y(x) / dx + d J (x) / dx y(x) I am not sure if this identity...
  9. 1

    Vector Calculus - Use of Identities

    Homework Statement By using a suitable vector identity for ∇ × (φA), where φ(r) is a scalar field and A(r) is a vector field, show that ∇ × (φ∇φ) = 0, where φ(r) is any scalar field. Homework Equations ∇×(φA) = (∇φ)×A+φ(∇×A)? The Attempt at a Solution I honestly have no idea how to even...
  10. D

    Vector identity proof using index notation

    Homework Statement I am trying to prove $$\vec{\nabla}(\vec{a}.\vec{b}) = (\vec{a}.\vec{\nabla})\vec{b} + (\vec{b}.\vec{\nabla})\vec{a} + \vec{b}\times\vec{\nabla}\times\vec{a} + \vec{a}\times\vec{\nabla}\times\vec{b}.$$ I can go from RHS to LHS by writng...
  11. C

    Prove this vector identity

    Homework Statement Show that: curl(r \times curlF)+(r.\nabla)curlF+2curlF=0, where r is a vector and F is a vector field. (Or letting G=curlF=\nabla \times F i.e. \nabla \times (r \times G) + (r.\nabla)G+2G=0) The Attempt at a Solution I used an identity to change it to reduce (?) it to...
  12. J

    Help with clifford algebra vector identity

    Homework Statement This is question 1.1 from section 2-1 of New Foundations of Classical Mechanics: Establish the following "vector identities": (a\wedge b) \cdot (c \wedge d) = b\cdot ca \cdot d - b\cdot da \cdot c = b\cdot(c\wedge d)\cdot a Homework Equations The Attempt at...
  13. K

    Vector Identity problem

    Homework Statement I want to compute the electric field knowing the magnetic field using a vector identity Homework Equations E=i \frac{c}{k} (∇\timesB) B(r,t)=(μ0ωk/4π) (\hat{r}×\vec{p})[1-\frac{1}{ikr}](eikr/r) \vec{p}=dipole moment,constant vector we have ti use the identity...
  14. A

    Vector Identity Proof

    Homework Statement Prove that: ∇×(a∙∇a) = a∙∇(∇×a) + (∇∙a)(∇×a) - (∇×a)∙∇a Homework Equations Related to the vorticity transport equation. The Attempt at a Solution Brand new to index/tensor notation, any suggestions on where to begin? For example, I am having trouble...
  15. A

    Proving the Vector Identity: a dot d(a)/dt = ||a|| x ||da/dt||

    Homework Statement Prove the following vector identity: Any vector a dotted with its time derivative is equal to the vector's scalar magnitude times the vector's derivative's scalar magnitude. Homework Equations (a)dot(d(a)/dt)=||a|| x ||da/dt|| The Attempt at a Solution I...
  16. R

    Vector Identity (del operator)

    i am completely lost as to how to go from p^ \frac{1}{m}∇p to \frac{m}{m+1} ∇p^\frac{m+1}{m}
  17. N

    An easy vector identity I can't prove

    Show the following, where U is a vector, and r is the position vector: \nabla(U.r) = U in polar coordinates Many Thanks
  18. C

    Vector identity involving grad and a function

    Homework Statement The question is to use index notation to show that the following is true, where a is a three-vector and f is some function. Homework Equations The Attempt at a Solution Hmmmm . . . I haven't really got anything to put here! I am starting to get to grips...
  19. X

    I do not understand this vector identity proof

    So I am trying to follow my professors notes. Here is my work on the proof. And on the bottom is my answer and his answer. I know my answer is wrong, as I do not fully understand how to convert the summations at the end to their vector quantities. Is my work incorrect...
  20. T

    General Relativity - Riemann Tensor and Killing Vector Identity

    Homework Statement I am trying to show that for a vector field Va which satisfies V_{a;b}+V_{b;a} that V_{a;b;c}=V_eR^e_{cba} using just the below identities. Homework Equations V_{a;b;c}-V_{a;c;b}=V_eR^e_{abc}(0) R^e_{abc}+R^e_{bca}+R^e_{cab}=0 (*) V_{a;b}+V_{b;a}=0 (**) The Attempt at a...
  21. A

    Proving vector identity using levi-civita tensor help

    Using the fact that we can write the vector cross-product in the form: (A× B)i =ε ijk Aj Bk , where εijk is the Levi-Civita tensor show that: ∇×( fA) = f ∇× A− A×∇f , where A is a vector function and f a scalar function. Could you please be as descriptive as possible; as I'm not sure...
  22. Saladsamurai

    Vector Identity Using Index Notation

    Homework Statement I am supposed to verify that \nabla\cdot(\mathbf{u}\times\mathbf{v}) = \mathbf{v}\cdot\nabla\times\mathbf{u} - \mathbf{u}\cdot\nabla\times\mathbf{v}\qquad(1)[/itex] I want to use index notation (and I think I am supposed to, though it does not say to explicitly) to...
  23. F

    Vector identity show that question

    Homework Statement u and v are vectors Homework Equations show that : mod(u x v)^2 +(u.v)^2 = mod(u)^2 x mod(v)^2 The Attempt at a Solution I thought about let u =(a,b,c) let v = (x,y,z) and then doing the calculations. However I have done this but then squaring everything out...
  24. M

    Understanding the Vector Identity and Its Matrix Representation

    vector identity?? Homework Statement The text that I'm reading has a line that reads \left(\mathbf{b}\mathbf{k}\cdot-\mathbf{b}\cdot\mathbf{k}\right)\mathbf{v}=\omega\mathbf{B} and I'm not sure what it means by \mathbf{b}\mathbf{k}; it's clearly not the dot product nor the cross product. A...
  25. B

    Vector identity proof using index notation

    Homework Statement Using index notation to prove \vec{\nabla}\times\left(\vec{A}\times\vec{B}\right) = \left(\vec{B}\bullet\vec{\nabla}\right)\vec{A} - \left(\vec{A}\bullet\vec{\nabla}\right)\vec{B} + \vec{A}\left(\vec{\nabla}\bullet\vec{B}\right) -...
  26. M

    Proof of Vector Identity Using Standard Identities | C^2 Scalar Functions

    Homework Statement Let f(x,y,z), g(x,y,z), h(x,y,z) be any C^2 scalar functions. Using the standard identities of vector analysis (provided in section 2 below), prove that \nabla \cdot ( f \nabla g \times \nabla h ) = \nabla f \cdot ( \nabla g \times \nabla h) Homework...
  27. M

    C^1 or C^2? Investigating Vector Identity

    One of the basic vector identities is \nabla \cdot (\nabla f \times \nabla g) = 0 Is this true if f and g are C^{1} ? (Or they must be C^{2} functions? Thanks!
  28. L

    Proving [b x c, c x a, a x b] = [a, b, c]^2 with Vector Identity Proof

    Homework Statement Hi. I need to prove that [b x c, c x a, a x b] = [a, b, c]2 for any three vectors a, b and c. Note that [a, b, c] = a(b x c)Homework Equations I tried using the identify (a x b) x c = (a.c)b - (a.b)c The Attempt at a Solution Using the above identity I got [b x c, c x a...
  29. Q

    Deriving a vector identity using Pauli spin matrices

    Homework Statement I'm supposed to derive the following: \left({\bf A} \cdot {\bf \sigma} \right) \left({\bf B }\cdot {\bf \sigma} \right) = {\bf A} \cdot {\bf B} I + i \left( {\bf A } \times {\bf B} \right) \cdot {\bf \sigma} using just the two following facts: Any 2x2 matrix can...
  30. G

    Vector Identity: Validity Checked

    Homework Statement This is a problem from a textbook, Riley Hobson and Bence 'Mathematical Methods for Physics and Engineering'. It asks to check the validity of a vector identity. If a, b and c are general vectors satisfying a x c = b x c, does this imply c . a - c . b = c|a-b| 2. The...
  31. F

    Expanding Vector Identity: ∆ x [(u.∆)u]

    Homework Statement Could someone please tell me how to expand: ∆ x [(u.∆)u] Homework Equations [b]3. The Attempt at a Solution thankyou
  32. O

    Vector Identity Rules Ax(BxC)

    Homework Statement Regarding the identity Ax(BxC) Homework Equations Does this identity only hold when A != B != C?
  33. E

    Proving a vector Identity

    Homework Statement Let the domain D be bounded by the surface S as in the divergence theorem, and assume that all fields satisfy the appropriate differentiability conditions. Suppose that: \nabla\cdot\vec{V}=0 \vec{W}=\nabla\phi with \phi = 0 on S prove...
  34. K

    Solving Vector Identity: {phi (grad phi)} X (n^)dS=0

    Homework Statement I am to show: closed integral {phi (grad phi)} X (n^)dS=0 Homework Equations The Attempt at a Solution I understand I am to use divergence theorem here.but cannot approach.Please help
  35. B

    Another vector identity question

    Hi, I'm stuck another vector identity question. It's of a different kind to the other one I asked about and looks so much easier but I just can't see what I need to do. I am told to use standard identities to deduce the following result. The standard identities being referred to are listed in...
  36. B

    A vector identity and surface integral

    Hi, can someone give me some assistance with the following questions? 1. Let f(x,y,z), g(x,y,z) and h(x,y,z) be any C^2 scalar functions. Prove that \nabla \bullet \left( {f\nabla g \times \nabla h} \right) = \nabla f \bullet \left( {\nabla g \times \nabla h} \right) . 2. Let S be the...
  37. Q

    Proving Vector Identity in Cartesian Coordinates

    Hello, I need some help on this vector identity. I am supposed to prove that Del Dot (Del(g(r)))=(2/r){dg(r)/dr}+(d^2g(r)/dr^2). Using Cartesian Coordinates. Any help would be GREATLY appreciated> :)
  38. Reshma

    What is the Vector Identity for \vec A and \hat n?

    Let \vec A be an arbitrary vector and let \hat n be a unit vector in some fixed direction. Show that \vec A = (\vec A .\hat n)\hat n + (\hat n \times \vec A)\times \hat n.
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