Vector analysis Definition and 126 Threads

Given a vector ##\mathbf{r} = r^i e_i## where ##r^i## are the components, ##e_i## are the basis vectors, and ##i = 1, \ldots, n##. In matrix notation, \begin{equation*} \mathbf{r} = \begin{bmatrix} e_1 & e_2 & \ldots e_n \end{bmatrix} \begin{bmatrix} r^1 \\ r^2 \\ \vdots\\ r^n \end{bmatrix}...

My book says, If the position vectors a, b, c of three points A,B,C and the scalars α, β, γ are such that αa + βb + γc=0,and α + β + γ = 0, then the three points A,B,C are collinear. On the other hand, If the position vectors a, b, c, d of the four points A,B,C,D (no three of which are...

I'm reading Group Theory by A. Zee , specifically, chapter I.3 on rotations. He used the passive transformation in analyzing a point ##P## in space. There are two observers, one labeled with unprimed coordinates and the other with primed coordinates. From the figure below, he deduced the...

I have seen conservative vector fields being defined as satisfying either of the two following conditions: The line integral of the vector field around a closed loop is zero. The line integral of the vector field along a path is the function of the endpoints of the curve. It is apparent to me...

##df=\frac {\partial f}{\partial r} dr+\frac {\partial f}{\partial \theta}d\theta\quad \nabla f=\frac{\partial f}{\partial r}\vec{e_r} +\frac{1}{r}\frac{\partial f}{\partial \theta }\vec{e_\theta }## On the other hand ## g_{rr}=1\:g_{r\theta}=0\:g_{\theta r}=0\;g_{\theta\theta}=r^2\;##So...

If we define Laplacian of scalar field in some curvilinear coordinates ## \Delta U## could we then just say what ##\Delta## is in that orthogonal coordinates and then act with the same operator on the vector field ## \Delta \vec{A}##?

I will only care about the ##t## and ##x## coordinates so that ##(t, z, x, x_i) \rightarrow (t,x)##. The normal vector is given by, ##n^\mu = g^{\mu\nu} \partial_\nu S ## How do I calculate ##n^\mu## in terms of ##U## given that the surface is written in terms of ##t## and ##x##? Also, after...

In many books it is just written that ##\Delta(\frac{1}{r})=0##. However it is only the case when ##r \neq 0##. In general case ##\Delta(\frac{1}{r})=-4\pi \delta(\vec{r})##. What abot ##\mbox{div}(\frac{\vec{r}}{r^3})##? What is that in case where we include also point ##0##?

In a certain anisotropic conductive material, the relationship between the current density ##\vec j## and the electric field ##\vec E## is given by: ##\vec j = \sigma_0\vec E + \sigma_1\vec n(\vec n\cdot\vec E)## where ##\vec n## is a constant unit vector. i) Calculate the angle between the...

Summary:: i) Set up a differential equation that describes how the pressure ##p## varies with the distance r from the center of the planet. Hint: You can base your reasoning on static equilibrium and Archimedes' principle. ii)Calculate how the atmospheric pressure p and the density of the...

How I would have guessed you were supposed to solve it: What you are supposed to do is just take the gradients of all the u:s and divide by the absolute value of the gradient? But what formula is that why is the way I did not the correct way to do it? Thanks in advance!

hi guys i saw this problem in my collage textbook on vector calculus , i don't know if the statement is wrong because it don't make sense to me so if anyone can help on getting a hint where to start i will appreciate it , basically it says : $$ \phi =\phi(\lambda x,\lambda y,\lambda...

Given $$\vec E = -\nabla \phi$$ there $$\vec d \rightarrow 0, \phi(\vec r) = \frac {\vec p \cdot \vec r} {r^3}$$ and ##\vec p## is the dipole moment defined as $$\vec p = q\vec d$$ It's quite trivial to show that ##\nabla \times \vec E = \nabla \times (-\nabla \phi) = 0##. However, I want to...

In Mathematical Methods for Physicists, Sixth Edition, Page 60, Section 1.11, the Gauss' theorem is written as: In Mathematical Methods for Physicists, Fifth Edition, Page 61, Section 1.11, the Gauss' theorem is written as: Kindly I would like to know please: 1. What is the difference between...

I wrote the equations of the Nabla, the divergence, the curl, and the Laplacian operators in cylindrical coordinates ##(ρ,φ,z)##. I was wondering how to define the direction of the unit vector ##\hat{φ}##. Can we obtain ##\hat{φ}## by evaluating the cross-product of ##\hat{ρ}## and ##\hat{z}##...

I just want to know the difference between those rules: 1. R^2 = F1^2 * F2^2 + 2*F1*F2*COS(the angle between F1 and F2) 2. The second is about the parallelogram rule, it says that the two vectors are added and their summation is the magnitude of the resultant. Which one is correct?

In Mathematical Methods for Physicists, 6th Edition, page 44, Example 1.8.2, the curl of the central force field is zero. 1. Why are central force fields irrotational? 2. Why are central force fields conservative? Any help is much appreciated...

I have a simple question about the notation of the nabla operator in Vector Analysis. The nabla operator is a vector differential operator and it is written as: $$\nabla = \hat{x} \frac {∂} {∂x} + \hat{y} \frac {∂} {∂y} + \hat{z} \frac {∂} {∂z}$$ Is it okay if we accented nabla by a right...

I have a question regarding the dot product and the cross product differentiation. I was wondering whether: $$\frac{d(\vec{A}.\vec{B})}{du} = \vec{A}. \frac{d\vec{B}}{du} + \frac{d\vec{A}}{du} .\vec{B}$$ is the same as $$\frac{d(\vec{A}.\vec{B})}{du} = \frac{d\vec{A}}{du} .\vec{B} + \vec{A}...

In Mathematical Methods for Physicists, 6th Edition, by Arfken and Weber, Chapter 1 Vector Analysis, pages 8-9, the authors make the following statement: "If Ax and Ay transform in the same way as x and y, the components of the general two-dimensional coordinate vector r, they are the...

Knowing that a scalar quantity doesn't change under rotation of a coordinate system. Do we consider a point in a Cartesian coordinate system (i.e. A (4,5)) a scalar quantity? If yes, why do the components of point A change under rotation of the coordinate system? According to my understanding...

Hi all, Not a question about completing homework here, but I'm a teacher looking to create a realistic engineering question for an assignment. I have an engineering scenario I've set for the assignment which is a junior engineer working for a marine engineering company so all of the questions...

Homework Statement A pilot wishes to fly at maximum speed due north. The plane can fly at 100km/h in still air. A 30km/h wind blows from the south-east. Calculate: a) The direction the plane must head to fly north. b) Its speed relative to the ground. Homework Equations Sine Rule...

This is more of a general question, but I've encountered this kind of exercises a lot in my current preperations for my exam: There are two cases but the excercise is pretty much the same: Compute $$(1) \space \operatorname{div}\vec{A}(\vec{r}) \qquad , where \thinspace...

Homework Statement Homework Equations Stokes theorem $$\int_C \textbf{F} . \textbf{dr} = \int_S \nabla \times \textbf{F} . \textbf{ds}$$ The Attempt at a Solution I have the answer to the problem but mine is missing a factor of$$\sqrt 2 $$ I can't seem to find my error

Homework Statement |a| = 2 |b| = ## \sqrt3## |a - 2b| = 2 Angle between a and bHomework EquationsThe Attempt at a Solution ##\theta## is angle between a and b So angle between a and -2b is 180-##\theta## [/B] ##|a-2b|^2## = |a|^2 + |2b|^2 -2|a||2b|cos(180-##\theta##) ##2^2## = 2^2 +...

Homework Statement Find a piecewise smooth parametric curve to the astroid. The astroid, given by $\phi(\theta) = (cos^3(\theta),sin^3(\theta))$, is not smooth, as we see singular points at 0, pi/2, 3pi/2, and 2pi. However is there a piecewise smooth curve? Homework Equations $\phi(\theta) =...

I studied the vector analysis in Arfken and Weber's textbook : Mathematical Methods for Physicists 5th edition. In this book they give the definition of vectors in N dimensions as the following: The set of ##N## quantities ##V_{j}## is said to be the components of an N-dimensional vector ##V##...

Homework Statement Equation (10.30) in Jackson is the first-order Born approximation. What is the second-order Born approximation? Homework EquationsThe Attempt at a Solution I can get the first-order Born approximation in Jackson's textbook. If I want to obtain the second-order (or n-th...

Why is it difficult to find really challenging vector analysis problems (problems about Green's, Stokes' and Gauss' theorems in a Calculus 3 course) in Calculus books? Most of the problems are elementary, at least that's the impression I have(I could be wrong). Is it really difficult to...

Books on multivariable calculus that I often see get good recommendations are, Vector Calculus, Linear Algebra, and Differential Forms by Hubbard and Hubbard Vector Calculus by Colley What are other good books with some material on differential forms like Hubbard and Colley? Books by Edwards...

I have to do a teaching assistant job on a multivariable calculus class, I have to survey books that can be useful as resources. Has anyone used this book by Bourne and Kendall? I noticed that the treatment of vector analysis seems good and the chapter on Cartesian tensors seem to be a good...

Homework Statement [/B] Let f and g be scalar functions of position. Show that: \nabla f \cdot \nabla(\nabla ^2 g)-\nabla g \cdot \nabla(\nabla ^2f) Can be written as the divergence of some vector function given in terms of f and g. Homework Equations [/B] All the identities given at...

I have seen and gone through this thread over and over again but still it is not clear. https://www.physicsforums.com/threads/vectors-one-forms-and-gradients.82943/The gradient in different coordinate systems is dependent on a metric But the 1-form is not dependent on a metric. It is a metric...

I have this problem with a complex integral and I'm having a lot of difficulty solving it: Show that for R and U both greater than 2a, \exists C > 0, independent of R,U,k and a, such that $$\int_{L_{-R,U}\cup L_{R,U}} \lvert f(z)\rvert\,\lvert dz\rvert \leqslant \frac{C}{kR}.$$ Where a > 0, k...

I'm currently an applied math major. I'm creating a schedule for my next semester and I have the choice to take either complex variables or vector analysis with linear algebra and a college geometry course(elective of choice), but I don't know which pairing will be less stressful. I am currently...

Homework Statement Can I, for all purposes, say that Nabla, on index notation, is $$\partial_i e_i$$ and treat it like a vector when calculating curl, divergence or gradient? For example, saying that $$\nabla \times \vec{V} = \partial_i \hat{e}_i \times V_j \hat{e}_j = \partial_i V_j (\hat{e}_i...

Vector fields confuses me. What are the differences between (##t## could be any variable, not just time): 1. If the position vector don't have an argument, ##\mathbf{r}=x\mathbf{\hat e}_x+y\mathbf{\hat e}_y+z\mathbf{\hat e}_z=(x,y,z)## so ##\mathbf{E}(\mathbf{r},t)=E_x(\mathbf{r},t)\mathbf{\hat...

I am trying to understand space vectors in 3 phase machines. If you have a balanced 3 phase system, the 3 phasors of voltage, current or whatever... should sum to 0. i<0 + i<-120 + i<-240 = 0. But in this image of a rotating space vector ...

Homework Statement I have the function: f(x,y)=x-y+2x^3/(x^2+y^2) when (x,y) is not equal to (0,0). Otherwise, f(x,y)=0. I need to find the partial derivatives at (0,0). With the use of the definition of the partial derivative as a limit, I get df/dx(0,0)=3 and df/dy(0,0)=-1. However, my...

This question really pertains to motivating why vectors have components whereas scalar functions do not, and why the components of a given vector transform under a coordinate transformation/ change of basis, while scalar functions transform trivially (i.e. ##\phi'(x')=\phi(x)##). In my more...

Hello from Italy I'm switching from CS to Physics BS because i personally find it more various and interesting (and in Rome there is one of the best physics school in the world). Mathematical analysis is a common subject and my credits will be recognised but i didn't study Linear algebra yet...

Homework Statement A line is given by the equation ##x + 2y - 3z = 7##. Find any vector in the direction parallel to this line in the Cartesian coordinate system. Homework Equations I imagine that there are some fundamental relationships I am missing here that would make this more...

what is electric current...a scaler or vector?? ...well I personally believe that it is somewhere in between the two extremes (what is not 0,need not be an 1 either :oldbiggrin: ) ...particularly because of the strange similarity we see in vector addition and phasor addition)...some people...

Hello all! I've just started to study general relativity and I'm a bit confused about dual basis vectors. If we have a vector space \textbf{V} and a basis \{\textbf{e}_i\}, I can define a dual basis \{\omega^i\} in \textbf{V}^* such that: \omega^i(\textbf{e}_j) = \delta^i_j But in some pdf and...

Hi there! It looks like you are trying to prove that the second derivatives of the magnetic potential function ##\mathbf{A}## belong to the class ##C(\mathbb{R}^3)##. This is a great question and involves some advanced mathematical techniques. One approach you can take is to use the dominated...

I am having trouble calculating if two objects with initial positions and velocity vectors will come within a given distance of one another and if so calculating where the closest approach is. Can anyone point me in the right direction? My initial thoughts are that both are linear functions...

Homework Statement A charge distribution has uniform density in the x-y directions and varies with z according to: ρ(z) = ρ0e−|z|/t where ρ0 and t are constants. (a) Find the potential V (z) and the electric field E(z) (b) Sketch them clearly showing their behaviors in the regions |z| ≪ t...

Hi, i now studying vector calculus, and for sheer curiosity i would like know if there exist a direct fashion to generalize the rotor operator, to more than 3 dimensions! On wiki there exist a voice https://en.wikipedia.org/wiki/Curl_(mathematics)#Generalizations , but I do not know how you...

I'm looking for good examples of physical motivation for integrals over scalar field. Here is an example I've found: If you want to know the final temperature of an object that travels through a medium described with a temperature field then you'll need a line integral It appears to me that...