Line integral Definition and 393 Threads

So, as i understand, the geometrical meaning of this type of integral should still be the area under the curve, however, I really do not see how you can obtain each infinitesimal rectangle from the dot product. I have understood the typical work example, that is, the line integral as the sum...

Homework Statement ##\int_\mathscr{C} \vec{F}(\vec{r})\cdot d\vec{r}; \vec{F}(x,y,z) = <sin z, cos \sqrt{y}, x^3>## I am assuming ##\vec{r}## is the usual ##\vec{c}## used, so maybe this is where I am incorrect The Attempt at a Solution C goes from (1,0,0) to (0,0,3) Parametrizing...

I have had some trouble with Kleppner and Kollenkow's derivation of work in a uniform force field. As the attached image shows, all three integrals (with respect to dx, dy, dz) are evaluated as follows: $$\int_{x_a, y_a, z_a} ^ {x_b, y_b, z_b}$$ . I am not sure how to proceed with such limits...

Why is preferable to use the line integral than the area integral over the complex plane?

Homework Statement Here is my problem : so far I've solved the line integral but I don't know what is the condition that must be met in order to be independant of the path given. I found the line integral to be: 27/28

Homework Statement Hello, I have a quick question about the following problem F = (2y+3)i+xzj+(yz-x)k and straight lines from (0,0,0) to (0,0,1) to (0,1,1) to (2,1,1) Considering C1 is the line from (0,0,0) to (0,0,1) C2 is the line from (0,0,1) to (0,1,1) and C3 is the line from (0,1,1) to...

Hi, I have a book that makes the equality. \vec{B}dV = (\vec{e_1}B_1 + \vec{e_2}B_2 + \vec{e_1}B_2)dx_1 dx_2 dx_3 \\[1ex] = dx_1 \vec{e}_1(B_1 dx_2 dx_3 ) + dx_2 \vec{e}_2(B_2 dx_1 dx_3 ) + dx_3 \vec{e}_3 (B_3 dx_1 dx_2) = (\vec{B}\cdot d\vec{S}) d\vec{l}. I'm a bit confused as to how it...

Watching this video http://youtu.be/1JnayXHhjlg?t=5m30s, I understood the ideia the Fourier transform, that is a continuous summation of sinusoids. But now If I have amplitude and phase as function of σ and ω, the summation wouldn't be ##\sum_\sigma \sum_\omega A_{\sigma \omega} \exp(i...

ok, my turn to ask a question. Problem: evaluate ∫_{C}xyds for x=t^2 and y = 2t from 0\leq t \leq 5 not sure what I did wrong, but here it goes: solve for ds: ds =\sqrt{(\frac{dx}{dt})^2+(\frac{dy}{dt})^2} = \sqrt{4t^2+4}=2\sqrt{t^2+1} substitute: ∫_{0}^5 4t^3\sqrt{t^2+1}dt...

Homework Statement Homework Equations Trued only 1st question.. Unfortunately I lost my notes about this and cannot find anything relevant to this. I think, ∫cF.dr = ∫cF.dr/dt dt .. also dr/dt isn't it = ∂x/∂ti +∂y/∂tj+∂z/∂tk Also it seems that C is with parabolic shape? Can...

Using my understanding of calculus, I don't understand why line integrals in 3-d space can give a result > 0. You are following a line and integrating under that line. The line has some length. But according to my understanding of calculus, it does not have a width. What is this arbitrary...

Homework Statement \int \vec{F} \cdot d\vec{r} where F=<y,0> and \vec{r}=unit circle. Homework Equations i'd prefer to do this one without greens theorem (using it is very easy). The Attempt at a Solution y=r\sin\theta and x=r\cos\theta. now \int \vec{F} \cdot d\vec{r}=\int...

Homework Statement Evaluate the line integral of F dot dr where f(x,y)=<3x^2,2x+y> and C is a straight line segment from (1,2) to (5,4) Homework Equations Unfortunately I was out with family obligations when we covered line integrals and surface integrals so am stuck with the textbook...

Homework Statement Integral closed line Integral ∫F ds where F = <y+sin(x^2), x^2 + e^y^2> and C is the circle of radius 4 centered at origin. Homework Equations The Attempt at a Solution so ds = c'(t)dt I believe... where c(t) = <4cos(t),4sin(t)> c'(t) = <-4sin(t),4cos(t)>...

(disregard the [5+5+5] in the question)attempt: dr=(et(cost)+(sint)et)\hat{i} + (-et(sint)+(cost)et)\hat{j} ∫<3+2xy, x2-3y2>\cdot<et(cost)+(sint)et, -et(sint)+(cost)et>dt ..at which point i remembered i had to parametrize F in terms of t, but didn't know how to do

Homework Statement Given a vector field F=-y/(x^2+y^2) i +x/(x^2 +y^2) Calculate the curl of it the line integral of it in a unit circle centered at O Homework Equations The Attempt at a Solution I calculated that the curl is 0 but the line integral is 2π. I don't think this...

Homework Statement Homework Equations The Attempt at a Solution $$\int _{ 0 }^{ 2\pi }{ xdx } \\ =-\int _{ 0 }^{ 2\pi }{ sintcostdt } \\ =0$$ It feels wrong.

Homework Statement We need to calculate this complex integral as line integral: Homework Equations The Attempt at a Solution This is correct, I guess: But not sure about this part: Are dx, dy, x, y chages correct or there is other method to use?

Homework Statement Say I have a line integral which I have simplified to: \int\int x+y dS Over some surface S, let's say 5x^2 + 3y^2 = 4 or something. Having arrived at this step, how do I determine dS? The formulas and methods we've been taught doesn't really lead to this step all...

Homework Statement Compute the line integral of \vec{v} = (rcos^{2}\theta)\widehat{r} - (rcos\theta sin\theta)\widehat{\theta} + 3r\widehat{\phi} over the line from (0,1,0) to (0,1,2) (in Cartesian coordinates) The Attempt at a Solution Well, I expressed the path as a...

hHomework Statement Evaluate ∫xy|dr| over the path given by x=t^3, y=t^2, t=0...2 Homework Equations x=t^3, y=t^2, t=0...2 The Attempt at a Solution x=t^3, y=t^2 y^(3/2) =x, y=t, x=t^(3/2), t=0...4 ∫0to4 t^5/2 [Sqrt((3t^(1/2))/2)^2 +(1)^2] =∫0to4 t^5/2 [Sqrt(9t/4 + 1) dt...

"Simple" Line Integral in Complex Numbers If anyone could please double-check my final result for this question it would be greatly appreciated. Rather than write out each step explicitly, I'll explain my approach and write out only the most important parts. "[E]valuate the given...

Homework Statement Evaluate ##\int_{(1,1)}^{(4,2)} (x + y)dx + (y - x)dy## along (a) the parabola y2 = x (b) a straight line (c) straight lines from (1,1) to (1,2) and then to (4,2) (d) the curve x = 2t2 + t + 1, y = t2 + 1 The Attempt at a Solution (a) is fine. For (b), I get...

Can someone tell me where my calculations are going wrong. I am integrating over C2: (Note Line integral over C1 and C3 are zero.) NOTE: The vector function f(x,y,z) that I am integrating over C2 is highlighted in red in the paint doc. The equation that I am using is: ∫[f (dot) unit...

Homework Statement Given a vector field F(x,y,z) = (yz + 3x^{2})\hat{i} + xz\hat{j} + xy\hat{k} Calculate the line integral ∫_{A}^{B}F\bullet dl where A = (0,1,3) and B = (1,2,2) Homework Equations Right, first of all, what is dl ? I've gone over all my course notes and...

I know that \oint_{C}\mathrm{d}\vec{l} = 0, for any closed curve C. But when i try to calculate the integral around the unit circle in polar coordinates, I get a result different from zero. Here is my approach : \oint_{C}\mathrm{d}\vec{l} = \int_{0}^{2\pi}\hat{\phi}\mathrm{d}\phi =...

Homework Statement Using line integrals, find the mass and the position of the center of mass of a thin wire in the shape of a half-circle x^{2} + y^{2} = r^{2}, x ≥ 0 and -r ≤ y≤ r if the linear density is ρ(x,y) = x^{2} + y^{2} The mass is given by the integral of the density along the...

Homework Statement "Consider the Vector field F(x,y)=<cos(sin(x)+y)cos(x)+e^x, cos(sin(x)+y)+y>. Compute the work done as you traverse the Archimedes spiral (r=θ) from (x,y)=(0,0) to (x,y)=(2∏,0). (Hint: check to see if the vector field is conservative) Homework Equations 1) F(x,y)=<P,Q>...

1. Consider the curve c= (x(t),y(t),z(t)) in space as t varies over [0, T ]. We could also parameterize this curve by c= x(τ^2 ),y(τ^2 ),z(τ^2) τ ∈ [0, sqrt(T)]. Show that one obtains the same value for the line integral using either parameterization. The line integral is just the integral for...

Homework Statement Find the mass of a wire in the shape of the parabola y=x2 for 1 \leq x\leq2 and with density p(x,y)=x. Homework Equations The Attempt at a Solution I just want to make sure I am setting this integral up right. Here is what I did: I parameterized the equation to x=t, y=t2...

Homework Statement hi I am trying to adjust this general integral to my problem, my problem consists of a semi-infinite rod, i.e. x in [0,∞) the primed variables are the integration variables Homework Equations http://img339.imageshack.us/img339/5038/42247711.jpg The Attempt at a...

Homework Statement Homework Equations I can't think of many to begin with. I've mainly been dealing with the simple forms of Cauchy's theorem so far, such as the Cauchy-Goursat theorem, and also Cauchy's integral formulas. However, these don't seem to have any direct implications here...

I want to verify I am doing this correctly first: Evaluate##\int_c (x^2ydx+xdy)## where the line is from (1,2) to (0.0) My method is different from the book, I am using vector value function method where ##<x(t),y(t)>-(x_0,y_0>=t\frac {d\vec r}{dt}## and ##\vec r=\hat x x(t)+\hat y y(t)##...

Homework Statement Homework Equations The Attempt at a Solution I used ∇ X F for part (a) and part (b) and found both to be ≠ 0. Thus both cases F is not conservative. I have no clue about the second part, as both arent conservative...

Homework Statement P is the part of the curve 9y²=4x³ between the points (1,-2/3) and (1,2/3). Evaluate the integral $$\int_P x ds $$ Homework Equations The Attempt at a Solution $$\int_P x ds = \int_P x |r'(t)| dt $$ My problem is that I cannot find a right...

1. Homework Statement Vector field is F=-y\hat{x} + x\hat{y} Compute the line integral along the path c(t)=( cos(t), sin(t) ) with 0≤t≤∏2. The attempt at a solution i started computing f.dl but how much is dl ? I took it dx\hat{x} +dy\hat{y} I'm not sure if using Cartesian coordinates is right ?

I want to integrate around a closed circular path on xy plane around the origin. Say the radius is b. So ##\oint d\vec l## where ##d\vec l=\hat{\phi}b d\phi## 1) If I just use polar( or spherical or even cylindrical) coordinates. R=b and \oint d\vec l\;=\;\hat{\phi}\int_0^{2\pi} b...

Hey, I'm studying for a physics degree and have a general curiosity about vector calculus. Having learned about surface and line integrals for scalar functions in multivariable calculus I've been having some issues translating them into vector calculus. Though conceptually I haven't had much...

Homework Statement don't know the line integral latex code but; \int\underline{r}\timesd\underline{r} from (a,0,0) to (a,0,2∏b) on the circular helix \underline{r} = (acos(λ), asin(λ), bλ) The Attempt at a Solution Its the multiple use of the position vector r in the question...

Use Stokes's theorem to show that line integral of ##\vec{F}(\vec{r})## over an curve ##L##, given by ##\int_L \vec{F}(\vec{r}) d\vec{r}##, depends only on the start and endpoint of ##L##, but not on the trajectory of ##L## between those two points. Hint: Consider two different curves, ##L##...

Homework Statement I don't understand the follow formula of the integral : Integral of ( E dot dL) from B to A What direction is the dr vector? Is it the direction of the line integral? Say I want to derive the formula for electric potential due to a point in Space. E has a direction vector...

Homework Statement \vec { F } \left( x,y \right) =u\left( x,y \right) \hat { i } +v\left( x,y \right) \hat { j } u\left( x,y \right) , v\left( x,y \right) are continuous on ℝ² \Gamma is piecewise smooth. Is \psi (x,y){ =\int { \vec { F } \left( x,y \right) \cdot \vec { dr } } }...

Homework Statement Let C be the semi-circle on the sphere x^2+y^2+z^2 = 2 from N = (0,0,\sqrt{2}) to S = (0,0, - \sqrt{2}) which passes through the point (1,1,0) Note that x=y for all (x,y,z) on C. Evaluate the integral : \int_C z^2dx + 2x^2dy +xydz Hint : Use as your parameter the angle θ...

Homework Statement Let C be the arc x=t^2, \space y=2t, \space z= \sqrt{4+3t} for t \in [-1,0] Evaluate the line integral : \int_{C} z^2dx + \sqrt{x}dy - 4xyz dz Homework Equations \int_{C} f(P) dx = \int_{a}^{b} f(P(t)) x'(t) dt for t \in [a,b] The Attempt at a Solution So...

Homework Statement Given by picture. Homework Equations W = F*dr The Attempt at a Solution Given by pictures.

Homework Statement Evaluate this line integral ∫ F . dr , where F = (3x2 sin y)i + (x3 cos y)j between the origin (0,0) and the point (2,4): (a) along straight line y = 2x (b) along curve y = x2 Homework Equations The Attempt at a Solution Part (a) dr = dx i + dy j ∫ [ (3x2 sin y) i + (x3...

This problem is about Line integral of Vector Field. I believe the equation i need to use is: \intF.dr = \intF.r'dt, with r = r(t) I try to solve it like this: C1: r1= < 1 - t , 3t , 0 > C2: r2= < 0 , 3 - 3t , t > C3: r3= < t , 0 , 1 - t > After some computation, I got stuck at the...

Homework Statement Please evaluate the line integral \oint dr\cdot\vec{v}, where \vec{v} = (y, 0, 0) along the curve C that is a square in the xy-plane of side length a center at \vec{r} = 0 a) by direct integration b) by Stokes' theoremHomework Equations Stokes' theorem: \oint V \cdot dr =...

Homework Statement For some scalar field f : U ⊆ Rn → R, the line integral along a piecewise smooth curve C ⊂ U is defined as \int_C f\, ds = \int_a^b f(\mathbf{r}(t)) |\mathbf{r}'(t)|\, dt where r: [a, b] → C is an arbitrary bijective parametrization of the curve C such that r(a) and r(b)...

Homework Statement Use Green's theorem to evaluate the line integral: ∫y3 dx + (x3 + 3xy2) dy where C is the path along the graph of y=x3 from (0,0) to (1,1) and from (1,1) to (0,0) along the graph of y=x. 2. The attempt at a solution I've completed two integrals for both paths (y=x3 &...