Questions tagged [line-integrals]

In mathematics, a line integral is an integral where the function to be integrated is evaluated along a curve. The terms path integral, curve integral, and curvilinear integral are also used.

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Line Integral - Newton's Law

Newton's law of gravitation says that the intensity of the gravitational force between two objects with masses M is given by the following expression: |F| = mMG / r² R is the distance between ...
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Finding the range of the parameter after parameterizing a line segment or a curve

I have these two planes: $x-y-z=0$ and $x+y+2z=o$ and I want to parameterize the line of intersection which is $x=3y$ to calculate the line integral from the origin to the point $(3,1,-2)$. $$\text{...
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Use greens theorem to find work done

Use Green's Theorem to find the work done by the force $\mathbf{F}(x,y)=x(x+y)\mathbf{i}+xy^2\mathbf{j}$ in moving a particle from the origin along the $x$-axis to $(1,0)$, then along the line segment ...
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172 views

Use Green’s Theorem to evaluate the line integral along the given positively oriented curve. (which approach to take)

Use Green’s Theorem to evaluate the line integral along the given positively oriented curve. $$\int_c y^3 \, dx - x^3 \, dy, C \text{ is the circle } x^2+y^2=4$$ Ok, so I'm not sure how to approach ...
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327 views

Evaluate the line integral by two methods: (a) directly and (b) using Green’s Theorem

Evaluate the line integral by two methods: (a) directly and (b) using Green’s Theorem So I thought I knew how to do this problem but when I did it directly, the areas I got for each line were 0+2/3+...
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600 views

Evaluating both sides of the Greens Theorem identity

Hi, above is the identity of Greens Theorem. Say $$F_1=3x-y$$ and $$F_2=x$$ and $R$ is the region bounded by $y=1+x$ and $y=(x-1)^2$ for $0 \leq x \leq 3$. I am trying to verify Green's Theorem for ...
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79 views

How to Calculate the line integral

Suppose $\vec{F}(x,y) = \displaystyle 6 \sin\left( \frac{x}{2} \right) \sin\left( \frac{y}{2} \right) \vec{i} - 6 \cos\left( \frac{x}{2} \right) \cos\left( \frac{y}{2} \right) \vec{j}$ and C is the ...
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Let $f(x,y) = \sqrt{8y+1}$ be a scalar valued function, and the curve $C$ described by $2x^2$ for $0<=x<=1$

What is the value of the line integral of $f(x,y)$? The formula that I was taught was line integral = integral (from a to b) f(r(t)||r'(t)||dt where r is the parameterization of the curve $C$ and ||...
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Show that the line integral is independent of path and evaluate the integral

The problem: 19-20 Show that the line integral is independent of path and evaluate the integral. 19. $\int_C 2xe^{-y}dx + (2y - x^2e^{-y})dy$, C is any path from $(1,0)$ to $(2,1)$ I ...
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why would $\int_{C}\vec{F}\cdot d\vec{r}$ be the same for 3 curves in a vector field?

The figure shows the vector field $F(x,y) = <2xy,x^2>$ and three curves that start at $(1,2)$ and end at $(3,2)$. why would $\int_{C}\vec{F}\cdot d\vec{r}$ be the same for 3 curves in a vector ...
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find a function$ f$ such that $\nabla f = F$

Determine whether or not $F$ is a conservative vector field. If it is, find a funcion $f$ such that $F=\nabla f$.$$F(x,y)=y^2e^{xy}\vec i+(1+xy)e^{xy}\vec j$$ I tried following the method in my book. ...
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How to calculate the length of the portion of curve from given conditions.

Let $l$ be the length of the portion of the curve $x=x(y)$ between the lines $y=1$ and $y=2$ where $x(y)$ staisfy $\sqrt \frac {1+y^2+y^4}{y} \ , x(1)=0$. Then find $l$ . The main thing I didn't get ...
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Integrate over a disk

In a proove that I have to do in DoCarmo Differential Forms and aplications, There is a step that I can't understand. If $D$ is a disk, $u(x,y);v(x,y)$ then: $$\int_{\partial D} \frac{u du+vdv}{u^2+...
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Evaluate the following line integral: $\int_l\sqrt{x^2+y^2}dl$

Evaluate the following line integral: $$\int_l\sqrt{x^2+y^2}dl$$ where $$l:x^2+y^2=ax$$ What I've already done is: $$x^2+y^2=ax \Rightarrow \left( x-\frac{a}{2} \right)^2+y^2=\left(\frac{a}{2}\right)...
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Proving $\oint\vec{F}(\vec{G}\cdot \hat{n}) d\sigma=\iiint[\vec{F}(\nabla \cdot \vec{G})+(\vec{G}\cdot\nabla)\vec{F}]dV$

Given $C^1$ vector fields $\vec{F}, \vec{G}$, show that: $$\unicode{x222F}_\Sigma\vec{F}(\vec{G}\cdot \hat{n}) d\sigma=\iiint_\Omega [\vec{F}(\nabla \cdot \vec{G})+(\vec{G}\cdot\nabla)\vec{F}]dV$$ I ...
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Find the line integral using the fundamental theorem of line integrals

$$\int_C (1+\cosh(y),x\sinh(y))d\vec{s}$$ Where C is a curve that goes from $(0,0)$ to $(1,1)$ I am not sure on how to proceed, I can find $\vec{F}=(1+\cosh(y),x\sinh(y))$ $\vec{f}:\nabla f=\vec{F}...
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How can I compute the partial derivative under the line integral sign if the integrand is non-conservative?

Consider the following integral $$I=\int_C^{\vec{y}_{0},\vec{y}} \vec{F}(\vec{x})\cdot d\vec{x}$$ where, for simplicity, $\vec{y}_{0}$ is some constant initial point on $C$ (which is an explicitly ...
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Find $\int_{C}Fdr$ for the given $F$ and $C$

I need to find $\int_{C}Fdr$. $$F = (2x-y+4)i + (5y+3x-6)j$$ $C$ is the triangle with vertices $(0,0)$, $(3,0)$, $(3,2)$ traversed counterclockwise. My first approach to doing this is splitting $C$ ...
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Use of polar coordinate delta function in line interation

Suppose constant value,I is defined as, \begin{align} I=\int_{circle}f(r,\theta)\ dl \end{align} where equation of the circle is defined as $r=\cos(\theta-\frac{\pi}{4})$ (see plot below). Now in an ...
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Find the line integral of $x^2+y^2$ over the polar curve $r=e^{\theta}$

Find the line integral of $x^2+y^2$ over the polar curve $r=e^{\theta}$ Not really sure on how to find the curve on terms of a parameter in order to evaluate the integral
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179 views

Path integral of the electric field from $(1,0)$ to $(0,1)$

Find the path integral of $E(x)=-kq(\frac{x}{(x^2+y^2)^\frac{3}{2}},\frac{y}{(x^2+y^2)^\frac{3}{2}})$ along the straight line connecting $(1,0)$ and $(0,1)$ by calculating its line integral. I really ...
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262 views

What is the intuition behind the line integral for the center of mass?

We define, for a density function $\rho(x,y)$, the following to be the center of mass for a wire defined by the curve $C$. $$\bar{x}= \frac{1}{M_\text{Total}}\int_Cx\rho(x,y)ds$$ $$\bar{y}= \frac{1}...
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Gramian determinant and line integral of a vector field

When we do a line integral over a scalar field, I understand this as a special case of an Integral over a one-dimensional real manifold. Getting the volume (in this case "length" as a 1-dim. volume) ...
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Evaluate $\int_c z^2\ dz$, where the curve $c$ is the line $x^2+y^2=\tan^{-1}(x,y)$

given the line $c$: $\Re(z)^2+\Im(z)^2=\arg(z),$ where $y$ is the imaginary part and $x$ is the real $$$$i want to evaluate $$\int_C z^2\ dz, z=x+iy$$ so: $$ \int_C z^2\ dz=\int_C\left(x+iy\right)^2\...
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318 views

Trying to prove the fundamental theorem for gradients.

I'm a beginner at vector calculus so I'm not really good at this yet. Trying to learn from the book I stated below: The fundamental theorem for gradients from the Vector Analysis section of Griffith'...
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variation of a smooth function on a curve

Assume $f$ is a smooth (e.g. $C^1$) real-valued function defined on a domain $\Delta\subset\mathbb{R}^n$. Let $t\in[a,b]\to\gamma(t)$ be a smooth curve in $\Delta$ with endpoints $x=\gamma(a)$ and $y=\...
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Proof of Goursat's theorem without using contradiction

In Tao's Notes 3 on complex analysis there is the following Exercise (number 10) Find a proof of Goursat's theorem that avoids explicit use of proof by contradiction. Goursat's theorem states that ...
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Calculate the line integral $I(a,b)=\int_{x^2+y^2=R^2}\limits\ln\frac{1}{\sqrt{(x-a)^2+(y-b)^2}} ds\quad(a^2+b^2\ne R^2).$

Calculate the line integral $$I(a,b)=\int_{x^2+y^2=R^2}\limits\ln\frac{1}{\sqrt{(x-a)^2+(y-b)^2}} ds\quad(a^2+b^2\ne R^2).$$ The parametrized integral path can be given as $$x=R\cos t,y=R\sin t,t\in[...
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Evaluate $\int_C (z^2,xz,2xy)\cdot dr$ where $C$ is the intersection of the surface $z=1-y^2, z\ge0$, and $2x+3z=6$ oriented anti-clockworkwise.

Evaluate the line integral of $\int_C F \cdot \,dr : F(x,y,z)=(z^2,xz,2xy)$ and $C$ is the curve obtained by the intersection of the surface $z=1-y^2$, $z \ge 0$, with the plane $2x+3z=6$, oriented ...
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436 views

Stokes' Theorem: surface integral over the lateral surface of a pyramid

Let $S$ be the lateral surface of the pyramid with points $(0,0,0)$, $(1,3,0)$, $(1,3,5)$ and $(0,3,0)$ as shown: Let $\mathbf{F}(x,y,z)=(y,2x,xyz)$ be a vector field. Evaluate the surface ...
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Stokes' theorem: find the radius that satisfy the line integral given

Let $C$ be a circunference of radius $a$ , in the plane $2x+2y+z=4$, centered at the point (1,2,-2). If $F(x,y,z)=(y-x,z-x,x-y)$, determine the value for $a$ such that $\oint_C F \cdot dr = -8\pi/...
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82 views

line integral of 1 with respect to omega

I am facing some trouble with understanding the solution to a line integral. Follows the integral: $$ \oint{d \Omega}= \int_0^{2\pi}{dl} \int_0^\pi \sin \theta ~ d\theta = 4\pi$$ I can easily ...
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741 views

2D Divergence Theorem: Question on the integral over the boundary curve

Let $\;F=(F_1,F_2)\;$ be a two-dimensional vector field and consider the rectangle $\;\mathcal R= PQRS\;$: If $\;\vec v\;$ is a function which gives outward-facing unit normal vectors to $\;\...
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Can a 1st differential equation have two different solutions?

I am given the following differential equation: $$x^2y'-y^2=1$$ where $y(1)=0$ and asked solve it: lets divide the DE by $x^2$, $$y'-\frac{1}{x^2}\cdot y^2=\frac{1}{x^2}, x\ne0$$ Now lets find the ...
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Is there an equivalent to the line integral for area in $\mathbb R^3$?

So the following line integral is known to give the area of some figure in $\mathbb R^2$: $$A=\frac{1}{2}\oint_C x\;dy-y\;dx$$ Is there an equivalent expression in $\mathbb R^3$ for volume?
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does this hold for any conservative vector field?

A vector field in n dimensions is path independent if for every output there is a surface of an n dimensional shell on which the output is the same. I think this is true intuitively because the line ...
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Evaluate the total mass of a wire

A wire has the shape of a curve obtained by the intersection of the portion of the sphere $x^2+y^2+z^2=4$, $y\geq 0$, with the plane $x+z=2$. Knowing that the density in each point of the wire is ...
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Two multivariable integral questions.

Let $\mathbf{F}(x, y) = (-y^2, xy)$ and $C = \Bigl\{\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 : y \ge 0 \Bigr\}$. Determine $\displaystyle \int_C \mathbf{F} \cdot d\mathbf{x}$ if $C$ is oriented counter ...
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When calculating work using line integrals how is it known that paramaterization will not change the path?

I am having a bit of an intuition problem here. As I am doing work problems on the work done by using line integrals it is said the work done for some but not all line integrals depends on the path. ...
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315 views

Line integral, not depend on orientation?

Let $A=(0,0)$ and $B=(0,1)$. Using $r_1:[0,1]\longrightarrow\mathbb{R}^2$. $r_1(t)=(0,1-t)$ $$\displaystyle\int_B^A 1=\int_0^11\,dt=1.$$ On the other hand, Using $r_2:[0,1]\longrightarrow\mathbb{R}...
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67 views

Line integral of the vector infinitesimal length of a line

I'm studying magnetic field and the mechanic actions that a vectorial field $\mathbf{B}$ impose to a circuit. Now, when it comes to the resulting force on a closed circuit (closed line) with a ...
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Evaluate Integral over conservative vector field

Evaluate the line integral $$ \int\limits_C F \cdot dr $$ where $\DeclareMathOperator{grad}{grad}F= \grad f$, $f(x,y,z)=\sin(x)\cos(y)\,z$ and $C$ is the circle $x^2 +y^2=1$ and $z=3$. I ...
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36 views

Is disc method derived from line integrals?

Last night I had the idea to validate the formula for volume of a cone using line integrals. The idea is to consider traversing the x-axis from the origin to some point $(h,0)$, where $h$ is the ...
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A Work-Kinetic energy proof that is mathematically sound?

I was looking at proofs of $W=\Delta \text{KE}$ along curves online and they all seem sloppy to me. I'm wondering if this specific proof is mathematically sound, or if anyone has seen a mathematically ...
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Common alternative to $C$ for line integrals?

I am aware that this is a soft-question (hence the tag) but feel this question still is warranted in this community. I'm writing a proof for the circumference of a circle using a line integral. ...
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Theorem about winding number of two closed paths [duplicate]

Let $\gamma_1, \gamma_2 : [0,1]\to\mathbb{C}$ closed paths and $w\in\mathbb{C}$ a point, so that $$\vert\gamma_2(t)-\gamma_1(t)\vert < \vert\gamma_1(t)-w\vert$$ on $[0,1]$. Prove that $$n(\gamma_1,...
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28 views

Square of a conservative field integrated over a closed curve is 0?

So we know that if $P$ is a conservative field, and $\tau$ is the unit tangent to closed curve $\Gamma$, then we have $\int_\Gamma P \cdot \tau ds = 0$ So my question is if $\int_\Gamma ( P \cdot \...
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149 views

Calculate $ \oint_C y^2\, dx + x\, dy $ using Green's Theorem?

Let $C$ be the curve parametrized by the equation $r(t) = 2\cos^3(t) i + 2\sin^3(t) j$ for $t \in [0,2\pi]$. I want to find the line integral $$ \oint_C y^2 \,dx + x \,dy . $$ I evaluated it ...
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123 views

Evaluate the line integral $\int_C zdx+xdy+ydz $

Calculate $\int_C zdx+xdy+ydz $ where $C= \{(x,y,z) \;| \;x=t, y=t^2, z=t^3, 0 \leq t \leq 1 \} $ So $$x=t $$ $$dx=1 $$ $$y=t^2 $$ $$dy=2t $$ $$z=t^3 $$ $$dz=3t^2 $$ $$\int_{0}^{1} [t^3(1)+t(2t)+t^...
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408 views

Find work done by the force field $F(x,y) = (3y^2 + 2)i + 16xj$ along the upper half of an ellipse

I'm trying to solve the following problem. (10.13.10 from Apostol Vol. 2.) Let $$ F(x,y) = (3y^2 + 2)i + 16xj $$ be a force field. I want to find the work done by $F(x,y)$ from $(-1,0)...