Identity Definition and 1000 Threads

Homework Statement Suppose that ##\langle S,*\rangle## has an identity e for *. If ##\phi : S \rightarrow S'## is an isomorphism of ##\langle S,*\rangle## with ##\langle S',*\rangle##, then ##\phi (e)## is an identity element for the binary operation ##*'## on S'. Homework EquationsThe Attempt...

Hi, I am working on an engineering problem and I have an equation which takes the following form: x = (A * cosα * sinθ) + (B * sinα * cosθ) Can this be further simplified? It almost looks like one of the sum-difference formulas you find in tables of trigonometric identities. I'm not to sure...

Homework Statement I have ##R^{u}_{o b u } F^{o}_a - R^{u}_{oau}F^{o}_{b}## and I want to show that this equal to ##2R_{o[aF^{o}_b]}## where ## [ ] ## denotes antisymmetrization , and ##F_{uv} ## is a anitymstric tensor Homework Equations Since ##F_{uv} ##is antisymetric the...

Homework Statement I have been going through a textbook trying to solve some of of these with somewhat formal proofs. This is a former Putnam exam question. (Seemingly the easiest one I have attempted, which worries me). Consider a polynomial function ƒ with real coefficients having the...

Homework Statement Homework Equations The Attempt at a Solution I am really lost here because our professor gave us no example problems leading up to the final exam and now we are expected to understand everything about vector calculus. This is my attempt at the cross product and...

Homework Statement The problem is to prove that ##A \cup (B - A) = \varnothing## Homework EquationsThe Attempt at a Solution The solution in the textbook is that ##A \cup (B-A) = \{x~ |~ x \in A \land (x \in B \land x \not\in A) \} = \{x~ |~ x \in A \land x \not\in A \land x \in B \} = \{x~ |...

Hi all, According to wikipedia: Can someone explain to me with a mathematical proof the following: $$ \frac {\partial f(x)} {\partial v} = \hat v \cdot \nabla f(x) $$ I don't get this identity except the special example where the partial derivative of f(x) wrt x is a special kind of a...

Homework Statement Problem 1: csc(tan^{-1}\dfrac{x}{2})=\sqrt{\dfrac{x^{2}+4}{x}} Problem 2: \sqrt{\dfrac{1-sinx}{1+sinx}}=\dfrac{|cosx|}{1+sinx} Homework Equations Quotient Identities tan\theta=\dfrac{sin\theta}{cos\theta} cos\theta=\dfrac{cos\theta}{sin\theta} Pythagorean Identites...

Hi, As part of showing that ##E^*_{2}(-1/t)=t^{2}E^*_{2}(t)## where ##E^*_{2}(t)= - \frac{3}{\pi I am (t) } + E_{2}(t) ## And since I have that ##t^{-2}E_{2}(-1/t)=E_{2}(t)+\frac{12}{2\pi i t} ## I conclude that I need to show that ##\frac{-1}{Im(-1/t)}+\frac{2t}{i} = \frac{-t^{2}}{Im(t)} ##...

I read an intuitive approach on this website. You should read it, it's worth it: https://betterexplained.com/articles/intuitive-understanding-of-eulers-formula/ I read that an imaginary exponent continuously rotates us perpendicularly, therefore, a circle is traced and we end up on -1 after...

Homework Statement $$sinx - cosx = 1/3$$ solve for $$sin(2x)$$ Homework Equations $$sin^2x + cos^2x = 1$$ $$sin2x = 2cosxsinx$$ The Attempt at a Solution I think you can square both sides and get: $$sin^2x - cos^2x = 1/9$$ But how can I use this information to solve for sin2x? Is there a...

I have a homework question and I am wondering if you can use Eulers identity in this case. If the equation is f(x)=constant*e^(-x^2) can this be rewritten as f(x)=consant*e^(ix)^2 and then, can you use the identity when it is in this form? Edit: Can it be put in the form cosx+isinx I am not...

Prove: ##(\cosh(x)+\sinh(x))^n=\cosh(nx)+\sinh(nx)## Newton's binomial: ##(a+b)^n=C^0_n a^n+C^1_n a^{n-1}b+...+C^n_n b^n## and: ##(a-b)^n~\rightarrow~(-1)^kC^k_n## I ignore the coefficients. $$(\cosh(x)+\sinh(x))^n=\cosh^n(x)+\cosh^{n-1}\sinh(x)+...+\sinh^n(x)$$...

Mod note: Reproduced contents of image with broken link: i = j x k j = k x i k = i x j Wikipedia says this about the standard basis vectors. Does this work for all (i.e, non basis) vectors? For example, if you know A = B X C does that mean C = A X B and B = C X A?

To me it seems basic question or even obvious but as I am not mathematician I would rather like to check. Is it true that these two matrices are both identity matrices: ##\begin{pmatrix}1&0\\0&1\end{pmatrix} ## and...

This is from a general relativity book but I think this is the appropriate location. The proof that \nabla_{[a} {R_{bc]d}}^e=0 is as follows: Choose coordinates such that \Gamma^a_{bc}=0 at an event. We have \nabla_a {R_{bcd}}^e = \partial_a \partial_b \Gamma^e_{cd} - \partial_a...

\tan\left({^2}\right)-\sin\left({^2}\right)=\tan\left({^2}\right) \sin\left({^2}\right) i keep on getting \sin\left({^2}\right)-\sin\left({^2}\right) \cos\left({^2}\right)=\sin\left({^2}\right) \sin\left({^2}\right) \cos\left({^2}\right)...

1 ___________ =csc2\theta-csc\thetacot\theta 1+cos\theta

Hello, sorry for the constant questions. But here is a question asking which of these are equal to the identity cot(x)/sin(2x). I managed to find out that this is equal to the third option of the three, however, apparently this option on its own is not the right answer. I can't seem to get the...

Before I start, there are only really two pieces of information this concerns and that is the idea that 1x = 1 and that ei*π = -1 So it would follow that (ei*π)i = -1i And so that would mean that i2i = e-π which doesn't seem to be right at all. Where is the issue here as there must be one but I...

I have encountered this equation: ##\cos^2 \gamma = \cos^2 \alpha \cdot \cos^2 \beta## According to the paper, this is a trigonometric identity, but this is the first time I have encountered this. The angles ##\alpha## and ##\beta## are somewhat similar to the components of the distance...

Homework Statement A solution of a trivalent metal ion is electrolysed by a current of 5.0A for 10 minutes during which time 1.18g of metal was plated out. The identity of the metal is: A cobalt B chromium C indium D gallium E bismuth Relative atomic masses: 1 faraday = 96,486 coulombs...

[mentor note] moved to homework forum hence no template. HI, I'm just having a bit of trouble with the numerator part of this identity ... Resolving the the denominator is fairly straightforward but .. Can anyone shed light on the final couple of steps (sin^3 x - cos^3 x)/(sinx + cosx) =...

The following identity is found in a book on Turbulence: Can someone provide a proof of this identity? It isn't listed in the list of vector calculus identities on Wiki. Thanks

Homework Statement Use index and comma notation to show: \begin{equation*}\text{div }(\text{curl } \underline{\bf{v}}) = 0\end{equation*} Homework Equations \begin{align*} & \text{(1) div } \underline{\bf{v}} = v_{i,i} \\ & \text{(2) curl } \underline{\bf{v}} = \epsilon_{ijk} v_{j,i}...

Homework Statement \text{Show that } \epsilon_{ijk} \epsilon_{mjk} = 2\delta_{im} Homework Equations \begin{equation*} \epsilon_{ijk} \epsilon_{mnp} = \left| \! \begin{array}{ccc} \delta_{im} & \delta_{in} & \delta_{ip} \\ \delta_{jm} & \delta_{jn} & \delta_{jp} \\ \delta_{km} &...

Hi! I have an integral to solve (that's not the point, though) and the inside of the integral is almost a trig identity: 1. Homework Statement ##sin\frac{(x+y)} {2}*cos\frac{(x-y)} {2} ## Homework Equations I noticed this was very similar to ##sinx+siny = 2sin \frac{(x+y)} {2} *...

Well, are there? I thought that problems involving the verification of identities pretty much checked themselves because you know whether the steps you’re doing are legitimate or not and, of course, you know whether you’ve reached the expression you want. However, I got one of these problems...

Hello! Please help me start solving this. I did expand the lhs but I still can't make it to be like the rhs. Any help would be appreciated!

Homework Statement ##tanx=\frac{(1+tan1)(1+tan2)-2}{(1-tan1)(1-tan2)-2}## find x Homework Equations 3. The Attempt at a Solution [/B] I tried multiplying through the paranthesis and arrived at ##tanx=\frac{(tan1tan2-1)+(tan2+tan1)}{(tan1tan2-1)-(tan2+tan1)}## and i don't know if this is any...

Hello everyone. I need help on this one Prove that $ \left((a-b)^2+(b-c)^2+(c-a)^2\right)^2=2\left((a-b)^4+(b-c)^4+(c-a)^4\right)$I noticed that the leftside of the eqn when expanded would be like $(X^2+Y^2+Z^2+2XY+2XZ+2YZ)$ from here I cannot move forward.

Let A= {1,2,3}. Let R= {<1,1>,<2,2>}. I(A) (Identity Realtion) on A >(def)> {<x,x>|x $$\in$$ A} So that mean : $$\forall$$ <x,x> x $$\in$$ A (That how I understood it) My question: Is R is identity relation on A ? Thank you !

Homework Statement Hi Everyone, Would somebody please be able able to check my working for the following problems: (Q1) Prove the identity of cos3θ - cos7θ/sin 7θ + sin3θ ≡ tan 2θ (Q2) Prove the identity of cos3θ ≡ 4 cos^3θ - 3 cos θ [hint: Express cos 3θ as cos (2θ+θ).] Homework...

Hey all. So I think I've been trying to figure this out longer than I should, that is why I am here now. I was reading an ieee paper and have been pondering a missing proof (the paper deems this proof too easy to show, literally... I must be really stupid or something). It is a simple question...

Homework Statement Here, V is a vector space. a) Show that identity element of addition is unique. b) If v, w and 0 belong to V and v + w = 0, then w = -v Homework EquationsThe Attempt at a Solution a) If u, 0', 0* belong to V, then u + 0' = u u + 0* = u Adding the additive inverse on both...

May someone kindly assist me to prove this trig identity (Sin2x-tanx) / cos2x = tanx Thank you for your assistance

For the given triangle as below: I can obtain trigonometry identities as below: ##\sin α = \frac{y}{r} = \cos (90° - α)## ##\cos α = \frac{x}{r} = \sin (90° - α)## ##\tan α = \frac{y}{x} = \cot (90° - α)## ##\cot α = \frac{x}{y} = \tan (90° - α)## ##\sec α = \frac{r}{x} = \csc (90° - α)##...

I have re-post this forum as I should have paid closer attention to rules. I apologized for that. Homework Statement 1) The expression tan^3 θ + sinθ/cosθ is equal to: (a) cot θ (b) tan θ sec^2 θ (c) tan θ (d) sin θ tan θ (e) tan θ csc^2 θ 2) Simplify (cos θ/1+ sin θ - cosθ/sinθ-1)^-1 (a)...

For the Gordon identity $$2m \bar{u}_{s'}(\textbf{p}')\gamma^{\mu}u_{s}(\textbf{p}) = \bar{u}_{s'}(\textbf{p}')[(p'+p)^{\mu} -2iS^{\mu\nu} (p'-p)_{\nu}]u_{s}(\textbf{p}) $$ If I plug in $\mu$=5, what exactly does the corresponding $(p'+p)^{5}$ represent? 4 vectors can only have 4 components so...

If $$\tan\left({\alpha}\right)\tan\left({\beta}\right)=1$$ $\alpha$ and $\beta$ are acute angles Then $$\sec\left({\alpha}\right)=\csc\left({\beta}\right)$$ Again there's options, I tried the product to sum formulas but it went off in a bad direction

If $\sin\left({\theta}\right)=\frac{\left(p-q\right)}{\left(p+q\right)}$ And $p$ and $q$ are $90^o<\theta<180^o$ and $p>q$ Show that $\tan\left({\theta}\right)=\frac{q-p}{2\sqrt{qp}}$ I tried using $q=\frac{2\pi}{3 }$ and $p=\frac{5\pi}{6}$ But not... To do this but theory I'm clueless

Hi everyone, I'm new to Physics Forums and to Mathematica, as well as Jacobi Identity. In any case, I was wondering on how I may use Mathematica to solve various Quantum Mechanics related problems through commutators. Like if it's possible to find out what is the form of a particular commutator...

Hey all, As I was working on my numerical PDEs homework, an identity came up which we used to solve a problem. I was able to answer the question, but my question here is where does the identity come from (I figured it has something to do with analysis) ? The identity is The integral of $$v^2$$...

I am currently reading Modern Electrodynamics by Andrew Zangwill and came across a section listing some delta function identities (Section 1.5.5 page 15 equation 1.122 for those interested), and there is one identity that really confused me. He states: \begin{align*} \frac{\partial}{\partial...

Hello! (Wave) I want to solve the following linear programming problem: $$\min (5y_1-10y_2+7y_3-3y_4) \\ y_1+y_2+7y_3+2y_4=3 \\ -2y_1-y_2+3y_3+3y_4=2 \\ 2y_1+2y_2+8y_3+y_4=4 \\ y_i \geq 0, i \in \{ 1, \dots, 4 \}$$ $\begin{bmatrix} 1 & 1 & 7 & 2 & | & 3\\ -2 & -1 & 3 & 3 & | & 2\\ 2 & 2 & 8...

Homework Statement Prove that \sum\limits_{k=0}^l{n \choose k}{m \choose l-k} = {n+m \choose k}Homework Equations Binomial theorem The Attempt at a Solution [/B] We know that (1+x)^n(1+x)^m = (1+x)^{n+m} which, by the binomial theorem, is equivalent to: {\sum\limits_{k=0}^n{n \choose...

Homework Statement Define n=(x + iy)/(2)½L and ñ=(x - iy)/(2)½L. Also, ∂n = L(∂x - i ∂y)/(2)½ and ∂ñ = L(∂x + i ∂y)/(2)½. with ∂n=∂/∂n, ∂x=∂/∂x, ∂y=∂/∂y, and L being the magnetic length. Show that a=(1/2)ñ+∂n and a†=(1/2)n -∂ñ a and a† are the lowering and raising operators of quantum...

As required by the Green's identity, the integrated function has to be smooth and continuous in the integration region Ω. How about if the function is just discontinuous at the boundary? For example, I intend to make a volume integration of a product of electric fields, the field function is...

I'm not sure I have the right approach here: Using the three 2 X 2 Pauli spin matrices, let $ \vec{\sigma} = \hat{x} \sigma_1 + \hat{y} \sigma_2 +\hat{z} \sigma_3 $ and $\vec{a}, \vec{b}$ are ordinary vectors, Show that $ \left( \vec{\sigma} \cdot \vec{a} \right) \left( \vec{\sigma} \cdot...

Homework Statement The antisymmetric tensor is constructed from a vector ##\vec a## according to ##A_{ij} = k\varepsilon_{ijk}a_k##. For which values of ##k## is ##A_{ij}A_{ij} = |\vec a|^2##? Homework Equations Identity ##\varepsilon_{ijk}\varepsilon_{klm} =...