Line integral Definition and 393 Threads

I'm getting a bit confused by the specific notation in the question and am unsure what exactly it is asking here/how to proceed. Homework Statement Given a scalar function ##f## find (a) ##∫f \vec {dl}## and (b) ##∫fdl## along a straight line from ##(0, 0, 0)## to ##(1, 1, 0)##.Homework...

Homework Statement Here are the three problems that i couldn't solve from the book Calculus volume 2 by apostol 10.9 Exercise 2. Find the amount of work done by the force f(x,y)=(x^2-y^2)i+2xyj in moving a particle (in a counter clockwise direction) once around the square bounded by the...

I am assuming that this line integral is along the straight line from $\displaystyle \begin{align*} (0,0,0) \end{align*}$ to $\displaystyle \begin{align*} \left( 5, \frac{1}{2}, \frac{\pi}{2} \right) \end{align*}$, which has equation $\displaystyle \begin{align*} \left( x, y, z \right) = t\left(...

Homework Statement A two dimensional force field f is give by the equation f(x,y)=cxyi+x^6 y^2j, where c is a positive constant. This force acts on a particle which must move from (0,0) to the line x=1 along a curve of the form y=ax^b where a>0 and b>0 Homework Equations Find a value of a(in...

Homework Statement Evaluate the following line integrals, showing your working. The path of integration in each case is anticlockwise around the four sides of the square OABC in the x−y plane whose edges are aligned with the coordinate axes. The length of each side of the square is a and one...

Homework Statement Let ##f(x,y)=(xy,y)## and ##\gamma:[0,2\pi]\rightarrowℝ^2##,##\gamma(t)=(r\cos(t),r\sin(t))##, ##r>0##. Calculate ##\int_\gamma{f{\cdot}d\gamma}##. Homework EquationsThe Attempt at a Solution The answer is 0. Here's my work. However, this method requires that you are...

Homework Statement I am having question with part c , for both c1 and c2 , here's my working for c1 , i didnt get the ans though . My ans is -5 , but the given ans for c1 and c2 is 27 , is the ans wrong ? Or which part i did wrongly ? Homework EquationsThe Attempt at a Solution

In our section on path independence, we were asked to find the potential function given a vector field. Our teacher says to use only line integrals to find the potential function, and not any other method. Like if we have ##F=\left< M,N,P \right> ## The first step is to determine if the domain...

Sorry where is pi/4 coming from in the line integral(section 3)? because i think it should be 1/2=tan(theta) which theta is 26.5651... it is impossible that the angle is pi/4? where is pi/4 coming from inside the circle? thank

Hello. I'm new to physics, and the problem I have seems so basic, mathematically speaking. I'm just failing to grasp exactly what is being asked. If I can find that, I believe I can find the answer. Here it is: 1. Homework Statement Let A = x2ˆx + y2ˆy + z2ˆz Consider the parabolic path y2 =...

I want to the line integral in the following picture: The field is the blue arrows that go left to right, and the path is the orange line that is going from right to left. Just by looking at the picture, it is clear that the result will be negative, but when I set up the integration this is...

Homework Statement Evaluate \int_C \vec{F} \cdot d\vec{r} Where \vec{F} is the field generated from a circular thread of radius b in the xy plane, with magnitude j in the direction \hat{\varphi} (i.e. not along the curve, I take it) C: (x,y,z) = b(1+ \cos{\alpha}, 0, \sin{\alpha}) 3. The...

Homework Statement for the line segment c2 , why did the author want to differentiate dx with respect to dy ? and he gt dx = 0 ? I'm curious why did he didnt do so for C3 , where dy= 0 ..Why didnt he also differentiate dy with dx ? dy/dx = 0 ? Homework EquationsThe Attempt at a Solution is...

Homework Statement i'm not sure what is line integral... Homework EquationsThe Attempt at a Solution Does it mean total length of line under the curve?

Homework Statement I need to find the work done by the force field: $$\vec{F}=(5x-8y\sqrt{x^2+y^2})\vec{i}+(4x+10y\sqrt{x^2+y^2})\vec{j}+z\vec{k}$$ moving a particle from a to b along a path given by: $$\vec{r}=\frac{1}{2}\cos(t)\vec{i}+\frac{1}{2}\sin(t)\vec{j}+4\arctan(t)\vec{k}$$ The Attempt...

This is part of a larger question, but this is the part I am having difficulty with. I have had an attempt, but am not sure where I am making a mistake. Any help would be very, very appreciated. 1. Homework Statement Let C2 be the part of an ellipse with centre at (4,0), horizontal semi-axis...

Homework Statement Evaluate ∫c (x + y) ds, where C is the circle centred at (1/2, 0) with radius 1/2. Homework EquationsThe Attempt at a Solution parametrise x=1/2cos(t) y=1/2sin(t) 0≤t≤2π ds=√dx2+dy2 =√(1/2)2-sin2(t)+(1/2)2cos2(t) =√-(1)2(1/2)2sin2(t)+(1/2)2cos2(t)...

Homework Statement Evaluate the length of the spiral with parametric equation ψ(t) =< 2 cost, 2 sin t, π/t >, with t ∈ [0, 2π]. Homework Equations Line integral ∫C f(x,y) dS The Attempt at a Solution f(x,y) = z = π/t ∫C π/t dS [0, 2π] are the lower and upper bounds of integration dS=...

Homework Statement Hi everybody! I would like to make sure I properly solved that problem because I find the result strange: Given a force field ##F_x = axy^3##, ##F_y = bx^2y^2##, ##F_z = cz^3##. Calculate the work with the line integral ##\int_{C} \vec{F} \cdot d\vec{r}## from point...

Fine the word done in moving a particle in the force field F=<2sin(x)cos(x), 0, 2z> along the path r=<t,t,t2>, 0≤t≤π To do the line integral, I need to find F(r(t)), but I don't understand how to express it. For example I looked at the online notes provided here...

Homework Statement Evaluate ∫C < −y, x − 1 > dr where C is the closed piecewise continuous curve formed by the line segment joining the point A(− √ 2, √ 2) to the point B( √ 2, − √ 2) followed by the arch of the circle of radius 2, centered at the origin, from B to A. 2. The attempt at a...

Homework Statement Use a line integral to find the area of the surface that extends upward from the semicircle ##y=\sqrt{9-x^2}## in the ##xy##-plane to the surface ##z=3x^4y## Homework Equations Parametric Equation for Circle: ## x = rcos(t) ## ## y = rsin(t) ## Line Integral: ## \int_c...

Homework Statement Need to visualize what it means by Line Integral along curve C with respect to x or y axis. For example suppose the curve is C (I did not find a way to write the C under the integration sign here) ∫ f(x,y) ds is like a fence along C whose height varies as per f(x,y). The...

Homework Statement Evaluate line integral(x+sqrt(y)) over y=x^2 from (0,0) to (1,1) and y=x from (1,1) to (0,0) Homework Equations n/a The Attempt at a Solution I set up integral from 0 to 1 of 2t(sqrt(1+4t^2))dt for the parabola part and then added integral from 0 to 1 of...

I'm used to parameterizing however I'm not sure how to solve these types of problems, any help would be much appreciated. 1) Calculate the line integral ∫v⋅dr along the curve y=x3 in the xy-plane when -1≤x≤2 and v=xyi+x2j 2) a) Find the work that the force F = (y2+5)i+(2xy-8)j carries...

Homework Statement \int xydx+ 4ydy where C is the curve from (1,2) to (3,5) made up of the twoline segments parallel to the coordinate axes. c_1:(1,2)\rightarrow(3,2) c_2:(3,2)\rightarrow(3,5) Homework EquationsThe Attempt at a Solution i got c2 correct, y=2+3t, and x = 0, for t goes from 0...

We just started going over line integrals in calculus, and have been told that the integral over any closed surface is 0. What I don't get is then why can we do the line integral of a circle to get 2##\pi##r? Since a circle is a closed surface, shouldn't the line integral then be 0?

Homework Statement The vector field ##\vec B## is given in spherical coordinates ##\vec B(r,\theta,\phi ) = \frac{B_0a}{r\sin \theta}\left( \sin \theta \hat r + \cos \theta \hat \theta + \hat \phi \right)##. Determine the line integral integral of ##\vec B## along the curve ##C## with the...

Homework Statement F[/B]=(y + yz- z, 5x+zx, 2y+xy ) use stokes on the line C that intersects: x^2 + y^2 + z^2 = 1 and y=1-x C is in the direction so that the positive direction in the point (1,0,0) is given by a vector (0,0,1) 2. The attempt at a solution I was thinking that I could decide...

Homework Statement A closed curve C is described by the following equations in a Cartesian coordinate system: where the parameter t runs monotonically from 0 to 2π, thus defining the direction of C. Calculate the area vector of the planar region enclosed by C, using the formula: 2. The...

What's the line integral of sphere? Is it possible to get the line integral in three dimensions? What kind of line are we integrating?

I just did this following exercise in my text If C is the line segment connecting the point (x_1,y_1) to (x_2,y_2), show that \int_C xdy - ydx = x_1y_2 - x_2y_1 I did, and I also noticed that if we put those points into a matrix with the first column (x_1,y_1) and the second column (x_2,y_2)...

This is an example at the beginning of the section on the Fundamental Theorem for Line Integrals. 1. Homework Statement Find the work done by the gravitational field \vec{F}(\vec{x}) = -\frac{mMG}{|\vec{x}|^3}\vec{x} in moving a particle from the point (3,4,12) to (2,2,0) along a piece wise...

Homework Statement Given the polar curve r = 3\sqrt{\cos{2\varphi}},\ -\frac{\pi}{4}\leq\varphi\leq\frac{\pi}{4}. Find the area of the surface enclosed by the curve using line integral of the second kind. Homework EquationsThe Attempt at a Solution According to Green's theorem: if F(x,y) and...

Can anyone please tell me where I am going wrong? I am getting the incorrect answer for the Word Done should be: WD = q(3x^2-6y) ... Apologies for not changing it into the format on here - but for my revision I have pretty much done that myself.

Homework Statement Use Green's Theorem to evaluate the line integral along the given positively oriented curve. ##\int_C y^4 dx + 2xy^3 dy ##, C is the ellipse ##x^2 + 2y^2 = 2##. Homework Equations Change of variables: ##\int \int_S f(x(u,v),y(u,v)) |{\frac {\partial(x,y)}{\partial (u,v)}}|...

Homework Statement Use Green's Theorem to evaluate the line integral along the given positively oriented curve. 1) Is the statement above the same as finding the area enclosed? 2) ##\int_C \cos ydx + x^2\sin ydy ##, C is the rectangle with vertices (0,0) (5,0) (5,2) and (0,2). 3) ##\int_C y^4...

Homework Statement Consider the vector field F(r) = Φ^ (a) Calculate ∫ F⋅dl where C is a circle of radius R (oriented counterclockwise) in the xy-plane centered on the origin. Homework Equations maybe Φ^ = -sinΦx^ + cosΦy^ The Attempt at a Solution not really a solution. i am just stuck at...

The question states: The calculation of the magnetic moment of a current loop leads to the line integral ∮ r x dr I am puzzled - shouldn't this be ∮ r x dl where r is the radius of the loop and dl is the small change along the loop? (I think dr would be in the same direction as r, so no cross...

Homework Statement Evaluate the line integral ∫^{(1,0)}_{(0,1)} (x^2-y)dx + (y^2+x)dy along (a) a straight line from (0,1) to (1,2); (b) the parabola x=t, y=t2 + 1; (c) straight lines from (0,1) to (1,1) and then from (1,1) to (1,2). Homework Equations Equation of line: y = mx + c The Attempt...

Homework Statement i have this problem and need your help. I tried to solve the first 2 question but don't know ho to solve the third one

Homework Statement First, let's take a look at the complex line integral. What is the geometry of the complex line integral? If we look at the real line integral GIF: [2]: http://en.wikipedia.org/wiki/File:Line_integral_of_scalar_field.gif The real line integral is a path, but then you...

Homework Statement Use Green's Theorum to evaluate the line integral ∫c (x^2)y dx, where c is the unit circle centered at the origin. Homework EquationsThe Attempt at a Solution Taking the partial derivative with respect to y and subtracting it from zero(I'm taking the dy in the original...

Homework Statement Find the line integral of f(x,y,z) = x+y+z over the straight-line segment from (1,2,3) to (0,-1,1). Homework Equations ∫ f(x,y,z)ds = ∫ f(g(t), h(t), k(t)) |v(t)| dt The Attempt at a Solution I arrived at the correct solution, but I'd like some clarity on the result...

Homework Statement Evaluate the line integral over the curve http://webwork.math.ttu.edu/webwork2_files/tmp/equations/33/92ca0e3b907f876e5a974ad1457d1f1.png from to http://webwork.math.ttu.edu/webwork2_files/tmp/equations/6f/bccf9dd59b9c22450c590a042bb77d1.png . ∫-ydx+3xdy (over the curve C)...

Homework Statement A force F has components F sub x = axy-by^2, F sub y= -axy+bx^2 where a= 2N/m^2 & b=4N/m^2 Calculate the work done on an object of mass 4kg if it is moved on a closed path from (x,y) values of (0,1) to (4,1), to (4,3) to (0,3) and back to (0,1). all coordinates are in metres...

Homework Statement Hello I was given the vector field: \vec A (\vec r) =(−y(x^2+y^2),x(x^2+y^2),xyz) and had to calculate the line integral \oint \vec A \cdot d \vec r over a circle centered at the origin in the xy-plane, with radius R , by using the theorem of Stokes. Another thing, when...

Hi everyone. I ran into a minor problem while trying to solve a problem on line integral. I suspect this question to be very straight-forward as it gave the parametric equations of the curve C. However, I am still unable to get the answer for some reason. May I have someone to point out where I...

Given the force (derived from a potential in planar polar coordinates) F(p,w) = 2p+sin(w)e_p+cos(w)e_w Where e_p and e_w are unit vectors How do I calculate the line integral over a circumference that is defined as: p = 2 0 ≤ w ≤ 2pi Using the definition of a line integral \int_0^{2pi} \...

Homework Statement Let F be a vector field F = [-y3, x3+e-y,0] The path in space is x2 + y2 = 25, z = 2. My parametrization is r(t) = [5cos(t),5sin(t),2] Homework Equations Line integral is the integral of F(r(t)) * r'(t) dt, where here the asterisk * is for the dot product, not normal...