Line integral Definition and 393 Threads

Homework Statement a. Find a parametric equation to describe a parabola from the point (1,1) to the point (2,4). b. Evaluate the line integral \int_C x ds along the parabolic segment in part a. Homework Equations \int_C x ds = \int_{t1}^{t2} x(t) |r'(t)| dt The Attempt at a Solution Well...

I am agonizing about the following integral identity: \frac{d}{dt} \int \int_{g(x,y) \leq t} f(x,y) dx dy = \int_{g(x,y)=t} f(x,y) \frac{1}{\left| \nabla g(x,y) \right|} ds, where ds is the line element. Clearly, using the Heavisite step function, the condition g(x,y) \leq t is...

Homework Statement Without parameterizing the path, determine what the value of the line integral (integral of F dot dr) is, if C is the closed, oriented path that travels around the triangle with vertices (0,0) (5,2), and (-3,6) and F=yi + xj Homework Equations Curl possiblY? The...

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Homework Statement for \varphi(x,y)=2x+y+10 ,calculate the flux line integral...on a straight line from A(1,4) to B(5,1). Homework Equations The Attempt at a Solution I tried to solve it but didnt get the right answer. first i found the quation of the line which i found to be...

Homework Statement A table of values of a function f with continuous gradient is given. Find the line integral over C of "gradient F dr" where C has parametric equations x = t2 + 1, y = t3 + t, 0<=t<= 1. Sorry, don't know latex. But here's a picture of the table and values...

http://img2.imageshack.us/img2/5061/14983795.jpg I have no idea how they simplified the integral to the second step.

Homework Statement Let f(x,y) and g(x,y) be continuously differentiable real-valued functions in a region R. Show that ∫f ∇g · dr ]= − ∫g ∇f · dr for any closed curve C in R. Homework Equations The Attempt at a Solution I don't really know where to start, so I tried to evaluate...

Homework Statement Evaluate the following line integral on the indicated curve C \int(y^2-x^2)ds C: x = 3t(1+t), y=t^3 ; 0 <= t <= 2 Homework Equations ds = \sqrt{(f'(t))^2+(g'(t))^2}dt The Attempt at a Solution dx/dt = 3+6t dy/dt = 3t^2 ds = \sqrt{(3+6t)^2+(3t^2)^2}dt ds =...

Homework Statement You are given a vector field A= kx2 x. a. First, calculate the line integral of A from x=-2 to x=2 along the x axis. b. Next, calculate the line integral of A between the same 2 points, but use a semicircular path with a center at the origin. Recall that in cylindrical...

So this is an ultra basic question, but I'm rusty with parametrization techniques and wanted to make sure I was doing this correctly. Let's say I want to evaluate \int_{\gamma} z \: dz where \gamma : [a,b]\rightarrow \mathbb{C} is some path of integration. Now, I figure I can parametrize the...

Homework Statement \int_{C}(xy+\ln{x})\mathrm{d}s where C is the arc of the parabola y=x^2 from (1,1) to (3,9) Homework Equations \int_{C}f(x,y)\mathrm{d}s = \int_{a}^{b}f(x(t),y(t))\sqrt{(\frac{dx}{dt})^2+(\frac{dy}{dt})^2}\mathrm{d}t The Attempt at a Solution Ok, so...

Homework Statement \ointxydx+x^2dy C is the rectangle with vertices (0,0),(0,1),(3,0), and (3,1) Evaluate the integral by two methods: (a) directly and (b) using green's theorem. Homework EquationsThe Attempt at a Solution Evaluating the integral directly: c1: y=0,x=t,dx=dt,dy=o...

The standard method of calculating area of ellipse: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 Area = \int_C -ydx \hbox { or } \int_C xdy It is more convient to use polar coordinate x=a cos \theta \; \hbox { and }\; y=b sin \theta dy = b cos \theta \hbox{ Using } \int_C xdy =...

Homework Statement [PLAIN]http://img843.imageshack.us/img843/3995/calc3.jpg The Attempt at a Solution Im here asking for some help in direction on how to do these problems so that I can find the solution by myself... I would really really appreciate any help anyone could provide...

Homework Statement hello again, sorry for asking so many questions, i just want to make sure if I am correct or not calculate the line integral y^2dx+x^2dy where line C is the triangle with sides x=1, y=0 and y=x The Attempt at a Solution first of all i tried to find a customization of the...

My course notes said that in greens theorem where the closed line integral of F.r = the double integral (...)dxdy the curve c is taken once anti-clockwise, why does it matter which way you take the line integral? Does it matter at all? Thanks

Homework Statement I'm attempting Q 3 from ch 16.4 of Stewart (p 1060). We are required to find the line integral where C is the triangle with vertices (0,0), (1,0) and (1,2). The line integral is Int xy dx + x^2*y^3 dy Homework Equations The Attempt at a Solution...

Homework Statement Calcualte the value of \int\limits_L \sqrt{x^2+y^2}dl, where L is an arc of a logarithmic spiral r=ae^{m\phi} between points A(0,a) and B(-\infty,0). Problem: I can't find a value of \phi where x=-\infty or y=a. Homework Equations We parametrise and get...

Homework Statement Calculate the line integral \int\limits_{AB} x^2 dx+ \sqrt{xy}dy , where AB is a part of a circle in the first quarter of carthesian coordinates system ranging from A(0,R) to b(R,0). Homework Equations The Attempt at a Solution Parametrisation of a circle...

The Integral I is defined by I = Integral F . dr Where F = (x-y, xy) << This is a verticle vector, i just didn'nt know how to write it with latex. And C is a triangle with the vertices (0,0), (1,0) and (1,3) tracked anticlockwise. Calculate the line integral using greens...

Homework Statement Integrate F(x) = x / |x|^3 along the straight line from (1,0,0) to (2,-2,1). Homework Equations Line integral = int (F dot dx) The Attempt at a Solution I don't know where to start. Usually I do line integrals by parameterizing the line and the vector field...

Can line integral of a vector field ever be zero? If can, what is the interpretation of this value (0) ? Thanks.

Hi everyone. I am going through examples for maths exams and am unsure on the final part of a question I am attempting so hoping you may help me? Homework Statement "Let C be the closed, piecewise smooth curve comprising individual curves C1 and C2 defined by r1 = (x, x2, 1) and r2 =...

I want to check my understanding of the line integral: For a scalar line integral, what we have geometrically is the area between a curve a given function, yes? Hence, it can be thought of as a kind of thin wall, correct? And where our function is f(x,y)=1, we have the length of the...

Apostol page 386, problem 5 Homework Statement Given f,g continuously differentiable on open connected S in the plane, show \oint_C{f\nabla g\cdot d\alpha}=-\oint_C{g\nabla f\cdot d\alpha} for any piecewise Jordan curve C. Homework Equations 1. Green's Theorem 2. \frac{\partial...

So, my multivariable class has just started line integrals, and I could use a little help with them. The problem I'm currently working on says: Evaluate the line integral, where C is the given curve: \int\limits_C \! xy \,ds C: x=t^2, y=2t, 0 \le t \le 1 I realize that, by eliminating the...

Homework Statement Compute the line integral of \intc ydx +zdy + xdz where c is the intersection of x^2 +y^2+z^2= 2(x+y) and x+y=2 (in the direction clockwise as viewed from the origin) Homework Equations The Attempt at a Solution While attempting this problem I had a few...

what is the different between line integral and surface integral? If we parameterize curve by x=t , y=t , what is the range of t ? Is it 0<= t <=1? why?

Homework Statement Homework Equations Given above. The Attempt at a Solution I attempted this problem first without looking at the hint. I've defined F(r) as (B+A)/2 + t(B-A)/2, with dr as (B-A)/2 dt . Thus F(r)dr = ((B+A)/2)*((B-A)/2)+((B-A)/2)^2 dt When I integrate this from -1 to 1 I...

Homework Statement Air is flowing with a speed of 0.4m/s in the direction of the vector (-1, -1, 1). Calculate the volume of air flowing per second through the loop which consists of straight lines joining, in turn, the following (1,1,0), (1,0,0), (0,0,0), (0,1,1), (1,1,1) and (1,1,0)...

Evaluate the line integral Force field is the integral in the form of integrand ( (2x dx+ 2y dy + 2 zdz)/r^2). the domain of integral is C, C = C1 + C2. C1 is the line segment from (1; 2; 5) to (2; 3; 3). C2,arc of the circle with radius 2 and centre (2; 3; 1) in the plane x = 2. The...

Homework Statement Evaluate \int[(3x-y)dx-xdy] where C consist of the parabola y=x^2 from (0,0) to (1,1) and then the line segment from (1,1) to (0,1) Homework Equations The Attempt at a Solution i did the integral of the y=x^2 parametrized x=t y=t^2 from 0 to 1 then i got my 1/2 but for the...

In some books I have seen: \oint \mathbf{E} \cdot d\mathbf{s}=0 Since the Electric Field is meant to be conservative. Elsewhere, however, I have also seen: \oint \mathbf{E} \cdot d\mathbf{s} = -\frac{d\Phi_B}{dt} What's going on here? Thanks

Can I consider the ordinary integral over the real line a special case of the line integral, where the line is straight and the field is defined only along the line?

In a line integral with respect to arc length, we have something like f(x, y)ds "inside" the integral sign. The ds tells us that we are working with the arc length function s, taking diferences (s_K+1 - s_k) in the sums that tend to the line integral. Question: do we shall understand that...

Homework Statement What is \int_{\gamma} xy dx + x^2 dy in each of the following cases? 1. \gamma is the lower half of the curve 2x^2 + 3y^2 = 8, traveled from (2,0) to (-2,0). 2. \gamma is the full curve 2x^2 + 3y^2 = 8, traveled counterclockwise. Homework Equations The line...

Hello, sorry for my English;D Homework Statement Can a vector field exist in polar/spherical system? is it possible to define line integral in these systems? does it make any sense a vector field defined in polar system, ex. \vec A\left(r,\varphi\right)=r^3? and a line integral from...

Homework Statement Prove that 2A=\oint \vec{r}\times d\vec{r} Homework Equations The Attempt at a Solution From stokes theorem we have \oint d\vec{r}\times \vec{r}=\int _{s}(d\vec{s}\times \nabla)\times \vec{r}= \int _{s}(2ds\frac{\partial f}{\partial x},-ds+ds\frac{\partial...

suppose i have a nonconservative vector field. and there is a path going from point A to point B. How do i determine the path taken from A to B such that the line integral is maximized? edit: actually after thinkin about it, this might be an undefined problem unless there is some constraint on...

Homework Statement Suppose C is the line segment from the point (1,0) to the point (3,1). Compute the line integral intC {( xdx + (x + y)}dy Homework Equations The Attempt at a Solution i graphed the line that connects(1,0) to (1,3) and i got the equation of that line so y...

Homework Statement What is the result of this? \ointr.dr=? Homework Equations The Attempt at a Solution \ointr.dr =\ointrdr=\int^{a}_{a}rdr = \frac{r^2}{2} \left| ^{a}_{a} = 0 Is it correct?

Homework Statement This is an example in my book, and this is the information in the question. Find the work done by thr force field F(x,y) = (1/2)xy[B] i + (1/4)x^2 j (with i and j vectors) on a particle that moves from (0,0) to (1,1) along each path (graph shows a x=y^2 curve from (0,0)...

Homework Statement An object weighing 1.2 pounds travels along a helix given by x=cost, y=sint, z=4t, 0<=t<=8pi. Find the work done by the object. Let's keep this in ft. Homework Equations g=32.174 ft/s2 f=m*g f=w*d The Attempt at a Solution r(t)=cos(t)i+sin(t)j+4(t)k I know I need an F...

so I have a semi circle that goes from \frac{5\pi}{4} to \frac{\pi}{4} so the angle between the x-axis the radius of the circle is 45 degrees. I have, letting the radius = a) \frac{1}{a} \int \.dl this is going to have an x and y component, I know the x component is 2a in the...

Homework Statement From the 1984 Ap Physics C Mechanics Exam: If a particle moves in such a way that its position is described as a function of time by x = t3/2, then its kinetic energy is proportional to: (a) t2 (b) t3/2 (c) t (d) t1/2 (e) t0 (i.e. kinetic energy is constant)...

First I want to greet everyone because I am new here. I have attended to applied electromagnetic course which seems to be pretty hard to understand and issues came up at very first time after I went at calculations. I try to explain this as good as possible. 1. Vectorfield F(x,y,z) =...

1.find the line integral of (x^3-y^3)dx +(x^3+y^3)dy over r, where r is the boundary of the region limited by x^2+y^2=1 and x^2+y^2=9 Homework Equations 3. i found that the line integral over the curve x^2+y^2=1 is 3*Pi/2 and the double integral of the region limited by...

Homework Statement calculate: \oint \frac{2-y}{x^2+(y-2)^2} dx + \frac{x}{x^2+(y-2)^2} dy where y = \sin{t} + 2, x = \cos{t}, 0 \leq t \leq \pi Homework Equations Green's Theorem. The Attempt at a Solution In what order should I do everything? I need to find the derivaties...

Homework Statement Find the value of the (surface) integral \int curl \textbf{A} \bullet \textbf{a} if the vector \textbf{A}=y \textbf{i}+z \textbf{j}+x \textbf{k} and S is the surface defined by the paraboloid z=1-x^2-y^2 Homework Equations x=s\cos\phi y=s\sin\phi...