Vector identity Definition and 38 Threads

$$\nabla \times B(r)=\frac{\mu _0}{4\pi} \int \nabla \times J(r') \times \frac{ (r-r')}{|r-r|^3}dV'$$ 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...

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...

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...

Homework Statement I'm trying to derive the vector identity: $$\nabla(\vec{A} \cdot \vec{B})$$Homework Equations $$ \nabla(\vec{A} \cdot \vec{B})=(\vec{B} \cdot \nabla) \vec{A} + ( \vec{A} \cdot \nabla ) \vec{B} + \vec{B} \times (\nabla \times \vec{A})+ \vec{A} \times ( \nabla \times \vec{B})$$...

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...

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...

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

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...

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...

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...

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...

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...

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...

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...

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...

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}

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

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...

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...

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...

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...

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...

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...

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...

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) -...

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...

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!

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...

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...

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...

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

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

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...

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

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...

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...

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> :)

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.