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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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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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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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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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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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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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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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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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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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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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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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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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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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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