What is Vector: Definition and 1000 Discussions

<Moderator's note: Image substituted by text.> 1. Homework Statement Given the following vector field, $$ \dfrac{2(x-1)\,dy - 2(y+1)\,dx}{(x-1)^2+(y+1)^2} $$ how do I integrate : The integral over the curve x^4 + y^4 = 1 x^4 + y^4 = 11 x^4 + y^4 = 21 x^4 + y^4 = 31 Homework Equations...

Homework Statement I have an exam on vectors and 2 dimensional motion today. I NEED to know if it is possible to solve any GRAPHICAL VECTOR PROBLEM using only INTEGRATION AND MATH WITHOUT ADDING VECTORS TOGETHER GRAPHICALLY. So given any word problem that is supposed to be solved using that...

Why do we not need to consider direction when determining the change in potential energy? Why do we need to consider it in case of force? Or am I interpreting the question correctly?

What is the covariant derivative of the position vector $\vec R$ in a general coordinate system? In which cases it is the same as the partial derivative ?

hi I am studying algebra and i have a question. why is important that something is a vector space?, i mean, what implications have? matrix, complex numbers , functions , n-tuples. What do these have in common, apart from being a vector space? why is so important that a certain set of...

Homework Statement I have to calculate the partial derivative of an arctan function. I have started to calculate it but I wonder if there is any simpler form, because if the simplest solution is this complex then it would make my further calculation pretty painful... Homework Equations $$\beta...

Hello, I am having trouble understanding a proof presented here: http://www.fen.bilkent.edu.tr/~ercelebi/Ax(BxC).pdf This is a proof of the triple product identity, but I don't understand the last step, where they calculate ##\lambda##. Don't you lose all generality when you state ##\vec A##...

On Riemannian manifolds ##\mathcal{M}## the covariant derivative can be used for parallel transport by using the Levi-Civita connection. That is Let ##\gamma(s)## be a smooth curve, and ##l_0 \in T_p\mathcal{M}## the tangent vector at ##\gamma(s_0)=p##. Then we can parallel transport ##l_0##...

Hello, I am learning about Excited states of Helium in my undergrad course. I was wondering if the total spin operator Ŝ is a vector quantity or not. Thanks for your help.

Homework Statement A plane, diving with constant speed at an angle of 53 degrees with the vertical, releases a projectile an altitude of 730m. The projectile hits the ground 5.00s after release. A plane, diving with constant speed at an angle of 53 degrees with the vertical, releases a...

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...

Homework Statement A child is flying a large kite of mass 6.8 kg on a windy day. At the moment shown the tension from the string on the kite has a magnitude of 17.0 N and makes an angle of = 32.6° from the vertical, and the acceleration of the kite has a magnitude of ak = 7.72 m/s2 and makes...

Hello, For calculating the mean power at a specific cross section of a waveguide, one can calculate the mean value of the temporal function of Poynting Vector, P(t), where P(t) is the ExHy-EyHx. Note that I am not talking about phasors or a sinusoidal state. If I integrate over the waveguide...

Homework Statement So I have these two Matrices: M = \begin{pmatrix} a & -a-b \\ 0 & a \\ \end{pmatrix} and N = \begin{pmatrix} c & 0 \\ d & -c \\ \end{pmatrix} Where a,b,c,d ∈ ℝ Find a base for M, N, M +N and M ∩ N. Homework Equations I know the 8 axioms about the vector spaces. The...

Homework Statement A sailboat sets out from the U.S. side of Lake Erie for a point on the Canadian side, 90.0 km due north. The sailor, however, ends up 50.0 km due east of the starting point. (a) How far, and (b) in what direction must the sailor now sail to reach the original destination...

Homework Statement In order to solve a question, I need to find the minimum and maximum resultant of three vectors, their magnitudes are given to me. Homework Equations Magnitude of vector A = 1 Magnitude of vector B = 3 Magnitude of vector A = 5 The Attempt at a Solution The maximum part was...

Can anyone help? What is the convention for inserting the vector symbol? I have a draft with -(\vec{2.5})^2 which displays correctly in preview ##-(\vec{2.5})^2\\## why is vec troublesome and underlined in red?? Should I ignore! I realize that I'm not using a variable.

Is there some easy way to see that [\vec{p}^2, \vec{p} \times \vec{L}] is equal zero? I use component method and got that.

In p.244 of Carroll's "Spacetime and Geometry," the Killing horizon ##\Sigma## of a Killing vector ##\chi## is defined by a null hypersurface on which ##\chi## is null. Then it says this ##\chi## is in fact normal to ## \Sigma## since a null surface cannot have two linearly independent null...

I read in several places that if, for example, a point particle exhibits uniform circular motion about the z-axis within an osculating plane not equal to the x,y plane, then the angular velocity still points along the z-axis, even though the angular momentum does not (it precesses about the...

Homework Statement Could someone explain how the property, $$\nabla (\frac{1}{R}) = -\frac{\hat{R}}{R^2}$$ where ##R## is the separation distance ##|\vec{r} - \vec{r'}|##, comes about? What does the expression ##\nabla (\frac{1}{R}) ## even mean? Homework EquationsThe Attempt at a Solution...

Homework Statement Could someone illustrate why $$\int_{V} \nabla \cdot (f\vec{A}) \ dv = \int_{V} f( \nabla \cdot \vec{A} ) \ dv + \int_{V} \vec{A} \cdot (\nabla f ) \ dv = \oint f\vec{A} \cdot \ d\vec{a}$$ ? Homework EquationsThe Attempt at a Solution I understand that the integrand can...

Homework Statement A -12nC charge is located at (x,y) = (1.0cm, 0cm). What are the electric fields at the positions (x,y) = (5.0cm, 0cm), (-5.0cm, 0cm), and (0cm, 5.0cm)? Write each electric field vector in component form. Homework Equations E=k(q/r2) The Attempt at a Solution I was able to...

My question is simply whether you can reduce a vector triple product, or more generally a scalar multiplier of a vector in a cross product? Given: (A x (uB x C) = v, where u and v are known constants. Is it valid to change that to: u(A x (B x C) = v or (A x uB) = v, can you change that to u(A...

Hi I am studying force components on a inclined plane. I understand the concept of breaking vectors into X , Y Components relative to the horizontal plane however what I can't seem to make sense of is how the ratios of a triangle and the main input force being the hypotenuse of the triangle...

1. Homework Statement Hi, I have done part a) by using the expression given for the lie derivative of a vector field and noting that if ##w## is a vector field then so is ##wf## and that was fine. In order to do part b) I need to use the expression given in the question but looking at a...

Homework Statement Use the relation ##\langle \vec e^a, \vec e_b \rangle = \delta^a_b## and the Leibniz rule to give an expression for the derivative of a dual frame vector ##\frac{\partial \vec e_b}{\partial x^a}## in terms of the connexion components. Homework EquationsThe Attempt at a...

Homework Statement https://i.imgur.com/UtLzb34.png Homework Equations the law of cosinus The Attempt at a Solution I have not been available to scan my work, but I'm kinda stuck at the beginning. And our teacher have not show us this kind of physics yet. Thanks.

Hello everyone, my question is: what are the criteria that must be satisfied for the pushforward of a smooth vector field to be a smooth vector field on its own right? Consider a smooth map \phi : M \longrightarrow N between the smooth manifolds M and N. The pushforward associated with this map...

Homework Statement My mentor has run me through the derivation of equation (3) bellow. I am unsure how he went from (1) to (3) by incorporating the log term from eq(2). In eq(3) it seems he just canceled the relevant n terms and then identified 1/n as the derivative of L however if this were...

Homework Statement A student pushes a 0.65kg box Ali g a desk. When he stops pushing the book, it moves 85cm before stoping (slowing down in this period). Coefficient of friction between book and Table is 0.27.Calculate the work done on the book by the friction. Should it be positive or...

I have seen two main different methods for finding the gradient of a vector from various websites but I'm not sure which one I should use or if the two are equivalent... The first method involves multiplying the gradient vector (del) by the vector in question to form a matrix. I believe the...

For one of my current projects in computer vision (which really is a study in point clustering and tracking in a data stream of n-dimensional data), I have come up with a way to very quickly index a 2D angle between two points or the angle of a vector. Doing a little bit of investigation, and...

Homework Statement Vector A + Vector B + Vector C = 0 A=B+C Homework Equations Find the angle between vector A and vector B The Attempt at a Solution I tried to solve this problem by resolving all the vectors and individually equating the value of vectors at each axes x, y and z to 0 but got...

Homework Statement A long straight wire of radius a and resistance per unit length R carries a constant current I. Find the Poynting vector N = E × H at the surface of the wire and give a sketch showing the directions of the current, the electric field E, the magnetic field H, and N. Integrate...

Homework Statement Consider the surface, S, in the xyz-space with the parametric representation: S: (, ) = [cos() , sin() , ] −1/2 ≤ ≤ 1/2 0 ≤ ≤ os(). The surface is placed in a fluid with the...

In a scientific paper (Neural Networks: A Systematic Introduction, page 86) about the Perceptron Algorithm I found: A good initial heuristic is to start with the average of the positive input vectors minus the average of the negative input vectors. In many cases this yields an initial vector...

Homework Statement At noon two boats P and Q have a position vector (i+7j)km and (3i+8j)km respectively to the origin O. i and j are unit vectors in direction to East and north respectively P is moving south east at 5√2 km/h and Q is moving at a constant velocity of (6i+5j) km/h At time t after...

Homework Statement Homework EquationsThe Attempt at a Solution I took ## \frac { d\vec r}{dt} = 0 ## . This gives ##t_m = \frac { 1} {\sqrt{ 2}} ## ...(1) To calculate ## \alpha ## , ## \cos{ \alpha } = \frac { \vec v \cdot \vec r}{|\vec v||\vec r|} ## ...(2) Then, magnitude of...

Homework Statement Let's say i was given Vector A and B. The Angle between them is 60 degrees. Vector A's magnitude is 40 and Vector B's magnitude is 50. Find magnitude of vector C, if C = vector A - vector B. Homework Equations I'm given the magnitudes, and need to find magnitude of C The...

I'm looking for a diagram or animation that shows the vector potential A (in the form of arrows or whatever) superimposed on the E and B fields of a plane EM wave. Since A is not unique, maybe two or three versions of the diagram (including one with Coulomb guage). An animation with a slider to...

Homework Statement Part C) Please: Homework Equations above,below The Attempt at a Solution so I think I understand the background of these expressions well enough, very briefly, changing the manifold from ## R^n ## to a cylindrical one- ##R^{(n-1)}^{+1}## we need to cater for winding...

Homework Statement A rotating magnetic dipole is built by two oscillating magnetic dipole moments, one along the y-axis and one along the x-axis. Find the vector potential at a point: (0, 0, ##z_0##) along the z-axis. Then find the magnetic field at ##z_0## . As the magnetic field is a function...

It makes me wonder... wikipedia says about a basis: In mathematics, a set of elements (vectors) in a vector space V is called a basis, or a set of basis vectors, if the vectors are linearly independent and every vector in the vector space is a linear combination of this set.[1] So what is the...

Homework Statement Three close-packed planes of atoms are stacked to form fcc lattice. The stacking sequence of the three planes can be altered to form the hexagonal close packed structure by sliding the third plane by the vector r over the second. If the planes in the fcc structure are all...

Homework Statement The velocity of a solid object rotating about an axis is a field \bar{v} (x,y,z) Show that \bar{\bigtriangledown }\times \bar{v} = 2\,\bar{\omega }, where \bar{\omega } is the angular velocity. Homework Equations 3. The Attempt at a Solution [/B] I tried to use the...

Homework Statement Example 2:[/B] Homework Equations Flux=integrate -Pgx-Qgy+R of the proj. area on xy plane for z=g(x,y) The Attempt at a Solution Why do my attempt is wrong? The example is using the foundational formula while I use the stock formula from the book, why is there a negative...

Homework Statement The stream function Ψ(x,y) = Asin(πnx)*sin(πmy) where m and n are consitive integers and A is a constant, describes circular flow in the region R = {(x,y): 0≤x≤1, 0≤y≤1 }. Graph several streamlines with A=10 and m=n=1 and describe the flow. Explain why the flow is confined to...

Homework Statement [/B] I would like to ask for Q5b function G & H Homework Equations answer: G: -2pi H: 0 by drawing the vector field The Attempt at a Solution the solution is like: by drawing the vector field, vector field of function G is always tangential to the circle in clockwise...

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) -...