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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question about this line integral whether it is exact

this form $\omega=-\frac y{x^2 + y^2}dx + \frac x{x^2 + y^2}dy$ ,which is not defined at origin .it is not exact on the whole x-y plane . but when y is not zero , $\omega=d(-\arctan(x/y))$, my ...
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38 views

Evaluate line integral for a vector field

Given the vector field $\mathbf{E}=\mathbf{a_x}y+\mathbf{a_y}x$, evaluate $\int\mathbf{E} \cdot \text{d}l$ from $P_3(3,4,-1)$ to $P_4(4,-3,-1)$ by converting both $\mathbf{E}$ and $P_3$ and $P_4$ into ...
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40 views

line integral of vector field on two circles

if c1 is a circle with radius 1 and origin of $\begin{bmatrix}1\\0\end{bmatrix}$ and c2 is a circle with radius 3 and origin of $\begin{bmatrix}0\\0\end{bmatrix}$ with counter counter clockwise ...
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31 views

Line integral of a continuous vector field

I'm studying line integrals of vector fields and I can't do this prove: Let $\gamma : [a,b] \rightarrow \mathbb{R}^m$ a regular parameterized curve e $F$ a continuous vector field defined over the ...
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Evaluate $\int_cydx+zdy+zdz$ if c is intersection of upper hemisphere $x^2+y^2+z^2=4$ $z \geq0$ and $x^2+y^2=2x$

Evaluate $\int_cydx+zdy+zdz$ if c is intersection of upper hemisphere $x^2+y^2+z^2=4$ $z \geq0$ and $x^2+y^2=2x$ oriented counter clockwise from xy plane First, I made the following parametricisation: ...
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Property of line integral of vector field

I have to prove the following $$ \left| \int_\Gamma F \cdot d\gamma \right| \leq M\cdot l(\Gamma) $$ where $M= \max\{ \|F(x)\|:x\in \Gamma \}$ and $l(\Gamma)$ is the lenght of $\Gamma$. I was using ...
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1answer
34 views

Interval of stokes theorem

I have the vectorfield $F=(y e^x,x^2+e^x,z^2)$ and the curve $r(t)=(1+cos(t),1+sin(t),1-cos(t)-sin(t))$ I have to find the line integral of this using stokes theorem, however in this case I'm only ...
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Geometric Interpretation of Conservative Vector Fields

I currently learning vector calculus and am having difficulty finding any auxiliary resources about the geometric interpretation of conservative vector fields. When learning the definition, and even ...
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213 views

Calculate $\int_0^{2\pi} \tan \frac{\theta}8 d\theta $ using complex analysis

Professor gave me the problem that calculates below real integral using complex analysis. $$\int_0^{2\pi} \tan \frac{\theta}8 d\theta $$ Actually this integral can easily be calculated just ...
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Other techniques for solving these ODEs?

I've been working on a line integral and calculus of variations problem where I am trying to minimize the line integral for a given scalar function $f(x,y)$. I have these two coupled, nonlinear, first-...
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How to determine the domain of the t parameter in line integrals?

So I started solving line integrals but I noticed that the line integral only specifies a condition, such as $x^2+y^2=1$, and then in the solvation there is suddenly specified a domain for the $t$ ...
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117 views

integral on a half circle

I was very confused by this integral: $I=\int_\gamma\frac{1}{z^2} dz$, where $\gamma$ is the upper half of the unit circle. I know that when $\gamma$ is the unit circle, this would integrate to 0. ...
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80 views

Line integral of non conservative vector field

I'm having trouble finding the line integral of this problem. I have been given a vector field $F=(2x\sin(\pi y)-e^z,\pi x^2\cos(\pi y)-3e^z,-xe^z)$ Where the curve $C$ intercepts between $z=\ln(1+x)$ ...
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Line and Surface Integral with the Dot Product replaced with a Cross Product

Having recently studied magnetostatics, I came across the Biot-Savart law, which is based on the line integal over a current distribution in a curve $C$: $$\mathbf B(\mathbf r)=\frac{\mu_0}{4\pi}\...
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2answers
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Residue Theorem for Gamma Function times Hurwitz Zeta function

I am trying to evaluate an integral which is a product of three functions: $$ \frac{1}{2\pi i}\int_{c-i\infty}^{c+i\infty} ds \,\left(\tilde{\beta}\sqrt{2r}\right)^{-s}\, \Gamma(s)\,\zeta_{H}\left(\...
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Prove that $|\int_{\gamma}\mathbf{F}\cdot \,ds | \leq LM$ where $L$ is the lenght of $\gamma$ and $\| \mathbf{F}(x,y,z)\| \leq M$

I need to solve this problem for my vector calculus class. Suppose $\gamma :[a,b]\rightarrow \mathbb{R}^3$ is $C^1$ path of length $L$ and $\mathbf{F}$ is a $C^1$ vector field in $\mathbb{R}^n$ with $...
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Finding the Path of a Piecewise Smooth Parametrized Curve

Suppose $\phi: [0,5] \to \mathbb{R}^2$ is the piecewise smooth parametrized curve given by \begin{align*} \phi(t) &= \begin{cases} \left(-1 + \cos\left(\frac{5\pi}{4} + \frac{\pi t}{2}\right), 1 +...
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4answers
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What does the notation for a line integral of a vector field actually mean?

I have been told that the line integral of a vector field, F(r) along a curve $C$ is: $$I =\int_C\textbf{F}\cdot \text{d}\textbf{r}=\int_C(F_x,F_y)\cdot (\text{d}x,\text{d}y),$$ where $\text{d}\textbf{...
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Evaluate Line integral with Green's Theorem

Calculate $$\displaystyle\oint_C(x\sin(e^y)+xy)dx+(\frac{x^2}{2}e^y\cos(e^y)+x^2y^3)dy$$ where $C$ is the polygon with vertices $(-1,0), (0,1), (1,1), (2,0), (0,-2) $ oriented counter clockwise. I ...
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If a line integral is 0, does the function have to be conservative?

If a line integral is 0, does the function have to be conservative? Take the line integral $\oint_{C}^{}y^4dx+2xy^3dy$, which is equal to 0 over the bounded region C: $x^2+2y^2=2$. However, I was ...
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37 views

Compute line integral $\int_a^b (y^2z^3dx + 2xyz^3dy + 3xy^2z^2dz)$

Compute line integral $\int_a^b (y^2z^3dx + 2xyz^3dy + 3xy^2z^2dz)$ where $a = (1,1,1)$ and $b = (2,2,2)$ What I have done: To find $t$ I used the calculation for slope: $\frac{x-1}{2-1}=t, \frac{y-1}{...
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Compute the line integral $\int_q xydx +(x^2+y^2)dy$

Compute the line integral $\int_q xydx +(x^2+y^2)dy$, where q is the first part in the first quadrant of a counterclockwise oriented circle $x^2+y^2=1$. When parameterized I get $f(x,y) = (\cos(t), \...
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Compute $\int_{C}\textbf{B}\cdot d\textbf{r} = 0$ directly C is a circle

I need to do a direct computation of $\int_{C}\textbf{B}\cdot d\textbf{r} $ where $\textbf{B} = (\frac{\mu_{0}I}{2\pi}\frac{-y}{x^2+y^2},\frac{\mu_{0}I}{2\pi}\frac{x}{x^2+y^2},0)$ and is a magnetic ...
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Compute $\int_{\gamma} \nabla f \cdot d\mathbf{x}$

Compute $\int_{\gamma} \nabla f \cdot d\mathbf{x}$ for the following choices of $f$ and $\gamma$. (a) $f(x,y) = x^2+y^2; \gamma:g(t) = (1+t^2, 1-t^2), -1 \le t \le2$ What I have tried: $$\int_{-1}^2(1+...
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1answer
29 views

Show equivalent parameterizations of line integral

Suppose that a curve $\gamma$ is parametrically defined by two continuously differentiable functions $f(t),$ $a \le t \le b$ and $g(u),$ $\alpha \le u \le \beta$. These functions are called equivalent ...
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about line integral evaluation

Any hints on which equation to use is most appreciated! I give up on thinking at 5am using my brain which is made up of mashed potatoes :) $\int\limits_C \underline{G} \cdot d\underline{r}$ with $G(x,...
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How to calculate the line integral of a vector field over a parabola

I am trying to answer a question about line integrals, I have had a go at it but I am not sure where I am supposed to incorporate the line integral into my solution. $$ \mathbf{V} = xy\hat{\mathbf{x}} ...
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Can a vector field be substituted as a vector in a formula?

Can this velocity vector field equation (which looks like this), be substituted into this formula's "u" or velocity vector? And would it give me this (assuming the other constants together ...
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How do you turn a velocity field to a force field for an airplane in a hurricane?

I'm quite new to this so please bear with me. I'm trying to simulate a plane going through a hurricane. To simulate this, I'm oversimplifying the hurricane into this velocity vector field (looks like ...
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1answer
30 views

Potential function for vector field $\frac{1}{r} \hat{\phi}$?

Question. Does the vector field $\vec{F}(r, \phi) = \frac{1}{r} \hat{\phi}$ have something like an associated potential function? Context. I know that $\vec{F}$ is not conservative, and so I should ...
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Find the value of k for which the line integral depends only on the coordinates of the end points of C.

∫_C[(1+ky^2)/(1+xy)^2 dx+(1+kx^2)/(1+xy)^2 dy] . Find the value of k for which the line integral depends only on the coordinates of the end points of C. Hence, for this value of k, determine the ...
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Evaluating $\int_{\gamma} (z^2 \sin^3z-(z-i)^3+2z+1)dz$

Note: we're not yet allowed to say derivative of holomorphic are holomorphic or even that they are continuous. hell. Question 1: Is this correct? For $\int_{\gamma} (z^2 \sin^3z-(z-i)^3+2z+1)dz$, ...
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Evaluating $\int_{\gamma}z^2e^{z^3}dz$

Note: we're not yet allowed to say derivative of holomorphic are holomorphic or even that they are continuous. hell. For $\int_{\gamma}z^2e^{z^3}dz$, where $\gamma:[0,1] \to \mathbb C$, $\gamma(t)=(t-...
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Setup complex line integral $\int_{\gamma}x^2+y^2 dz$, where $\gamma$ is the line segment from $i+2$ to $-i-1$

Question 1. Is this correct? I have to evaluate $\int_{\gamma}x^2+y^2 dz$, where $\gamma$ is the line segment from $i+2$ to $-i-1$ Step 1: Equation of line passing through $(2,1)$ and $(-1,-1) $is $...
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Prove $|e^{z^2+1}| \le e^2$

Actually what I want to prove is $|\int_{D} e^{z^2+1} dz| \le 2\pi e^2$ for $D$ = the unit circle (centred at origin). What I know is $|\int_\gamma f(z)dz| \le AB$ for $A$ the length of $\gamma$ (for ...
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How to show that a one-form is closed

For $c \colon [0, 2π] \to \mathbb{R}^3$ with the helix $c(t) = (\cos t,\sin t, t)$ and the 1-form on $\mathbb{R}^3$ $$\alpha = 2x_1 x_2 \,dx_1+ (x_1)^2\, dx_2 + x_3\,dx_3$$ How do you find the ...
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1answer
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Line integral with Green theorem

I will compute $\int_C \ e^xdx+xydy$ where C is the triangle with vertices (0,0), (1,1) and (0,2) with a positive orientation. I started with $\iint (\frac{\partial Q}{\partial x}-\frac{\partial P}{\...
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Is my intuition of line integral or curl wrong?

My impression was that the curl of a vector field measures how fast a vector field turns along a closed curve around that point. Consider the vector fields $\vec{V}_1=-y\hat{i}+x\hat{j}$ and $\vec{V}...
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Discretising a line integral

UPDATE 2: This question is a work in progress. Either the paper I was referring to is wrong, or I made a mistake translating what's in section 3.2 of the paper into this question (e.g. ...
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1answer
42 views

Path independence of integral of 1-form implies the form is exact

I am trying to prove that path independence of integral of 1-form implies the form is exact. Suppose we have 1-form $\omega=Pdx+Qdy$. We need to show that there exist function $F: \mathbb{R}^2 \supset ...
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Apply Generalized Stokes to $R^2$ Line Integrals

Is there a simple way to apply generalize stokes theorem to $\int_{\partial C} f(x,y) ds$ for some $C\subset R^2$. I am stuck on what $\omega$ would be in the formula.
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How can we calculate this line integral?

I am currently working on the problem given below. I tried to solve it, but I can't find a proper solution. We need to calculate the given integral of the function f(x,y,z) from pi to 2.5pi, along ...
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1answer
54 views

Calculating a line integral along a curve

I am struggling with the following question; how can I calculate the line integral $\int_C f(r) \, dr$, where $C$ starts at the point $(0,0,0)^\top$, and ends at the point $(a,a,a)^\top$ along the ...
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45 views

When is the gradient of a line integral equal to the vector field being integrated

Let $\newcommand{\R}{\mathbb{R}}$ $g:\R^n \to \R^n$ be a $C^\infty$ vector field. We know that if $g$ is conservative (or exact as a 1-form) then the gradient theorem (stokes theorem) tells us $$ g = \...
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1answer
37 views

Question about orientation of a line integral of 2 points of a circle

Calculate $$\int_C \frac{x^2-y^2}{x^2+y^2}ds$$ where C is the circle $x^2 + y^2 = 4$ from $A = ({2,0})$ to $B = (-1, \sqrt{3})$ I calculated $\,\,\,\,r = (2\cos{t},2\sin{t})\,\,\,\,$ and $\,\,\,\,||r'(...
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52 views

Dot product in a line integral of a vector field

I struggle to understand the meaning of the dot product in the line integral of a vector field. $$ W = \int_C \vec F \cdot d \vec r = \int_a^b F(\vec r(t)) \cdot r'(t) \ dt $$ The right-hand side is ...
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2answers
99 views

Integrate $x+y$ over the triangle with the vertices $(0,0),(1,0),(0,1)$

Integrate $f(x,y) = x+y$ along the triangle with the vertices $(0,0),(1,0),(0,1)$ I have set $\gamma_{trinagle}=\gamma_1+\gamma_2+\gamma_3$ where: $$\gamma_1(t)=(t,0), t\in[0,1]$$ $$\gamma_2(t)=(1-t,...
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52 views

Inequality for Complex line integral

I want to prove the following inequality, where $\gamma$ is a complex path with $[\gamma] \neq0$ and $f:[\gamma]\to \Bbb C$ a continuous function $$\biggl\lvert\int_\gamma \frac {f(z)}zdz\biggr\rvert^...
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1answer
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Problem in understanding the evaluation of an improper integral using residue theorem.

Show that $\displaystyle {\int_{0}^{\infty} \frac {x^{-c}} {1+x}\ dx = \frac {\pi} {\sin \pi c}}$ if $0 \lt c \lt 1.$ I have found this example in Conway's book on complex analysis in page no. $119.$ ...
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With what force $M$ distributed with uniform density over the circle $x^2 + y^2 = a^2$, $z = 0$ act on $m$ located at the point $A(0,0,b)$.

With what force will a mass $M$ distributed with uniform density over the circle $x^2 + y^2 = a^2$, $z = 0$ act on a mass $m$ located at the point $A(0,0,b)$. I want to setup the problem as a line ...

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