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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.

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; contour integral as well, although that is typically reserved for line integrals in the complex plane.

The function to be integrated may be a scalar field or a vector field. The value of the line integral is the sum of values of the field at all points on the curve, weighted by some scalar function on the curve (commonly arc length or, for a vector field, the scalar product of the vector field with a differential vector in the curve).

Read more on wikipedia's entry Line integral.

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Interpreting Line Integrals with respect to $x$ or $y$

A line integral (with respect to arc length) can be interpreted geometrically as the area under $f(x,y)$ along $C$ as in the picture. You sum up the areas of all the infinitesimally small 'rectangles' formed by $f(x,y)$ and $ds$. What I'm wondering…
Jim_CS
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Difference between Stokes' Theorem and Divergence Theorem

Recently learnt the two and I really can't tell the difference. I'm not sure if I'm missing something, but it really seems to me that they evaluate the same thing just using different methods. With Stokes' Theorem, it seems to me that we evaluate…
hs_math_enjoyer
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Vector fields, line integrals and surface integrals - Why one measures flux across the boundary and the other along?

Why is it that a line integral of a vector field takes the dot product of the vector field with the tangent? This results in us taking the component of the vector field in the direction of the tangent of the curve we are integrating over. This can…
14tim4
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Line integral with respect to arc length

I ran into a problem that, initially, I thought was a typo. $$\int_C\ e^xdx $$ where C is the arc of the curve $x=y^3$ from $(-1,-1)$ to $(1,1)$. I have only encountered line integrals with $ds$ before, not $dx$ (or $dy$, for that matter). At first,…
Somatic
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How can I prove that these definitions of curl are equivalent?

I am reading the book "Div, Grad, Curl, and All that" and I got to the section about curl. In this section, the author defines the curl to be $$ (\nabla \times \mathbf{F})\cdot \mathbf{\hat{n}} \ \overset{\underset{\mathrm{def}}{}}{=} \lim_{S \to…
Robert Lee
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When is the line integral independent of parameterization?

Let $\alpha: [a,b] \rightarrow \mathbb{R}^2$ be a smooth path (i.e. $\alpha'$ is continuous on $[a,b]$), and let $f$ be a continuous vector field. The line integral of $f$ along $\alpha$ is defined as $\int_a^b f[\alpha(t)]\cdot \alpha'(t) dt$. In a…
nowhere
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Evaluate the line integral along a parabola

Evaluate the line integral of $f(x,y)=-y+x$ along part of the parabola $y=2(x+1)^2$ from the point $(0,2)$ to the point $(-1,0)$ I need help trying to find a good parameterization for this because what I've done just lands me in a mess. My work so…
user8290579
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Line integral over conservative vector field not independent of path?

I have a vector field $$F(x,y) = \langle x^2 + y^2, 2xy\rangle$$ which is conservative. I also have a curve $C$ defined as the boundary of the region between the two curves $y = x+2$ and $y = x^2$. Since $F$ is conservative, I know that the…
TheAssistant
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Conservativeness of $e^z$

So I recently have started trying to dive deeper into the realm of complex analysis, but to get there I needed some notions of multivariable/vector calculus for stuff like line integrals, etc... I don't own a book or anything to follow, and instead…
Lucas Hallal
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Find Surface Area Via a Line Integral (Stokes' Theorem)

I am trying to use Stokes' Theorem to calculate the surface area of a parametrized surface via a line integral. The surface is the part of $z= x^2+y^2$ below the plane $z=5$. I know this can be done the usual way, without Stokes' Theorem, but there…
eg001
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Why are line integrals not always path independent?

The text I am reading says that a line integral, $$\int_{C}{\mathbf{F}\cdot\textrm{ d}\mathbf{r}}$$ is path-independent whenever $\mathbf{F}$ is a gradient field (or in the realm of physics, a conservative field). My question is, when is a line…
Andrew Paul
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Using Green's Theorem to Express the Integral $I=\int_C (Pdx+Qdy)$ as an expression of $I_i=\int _{C_i} (Pdx+Qdy)$

Let $p_1,...p_n$ be points in $\mathbb{R}^n$. Let $P(x,y), Q(x,y)$ be functions with continuous derivatives in $ D=\mathbb{R}^2\setminus\{p_1,...p_n\}$ such that $Q_x-P_y=1$ for all $(x,y)\in D$. For all $i$ let $C_i$ be a circle of radius $r_i$…
AnonymouseCat
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What is a geometric meaning of $2\pi i$ for complex integration?

Cauchy Integral formula Let $G$ be open in $\mathbb{C}$ and $f$ be holomorphic on $G$. Let $\gamma$ be a closed rectifiable curve in $G$ such that $Wnd(\gamma,z)=0$ for all $z\in \mathbb{C}\setminus G$. Then, $Wnd(\gamma,z) f(z)= \frac{1}{2\pi…
Rubertos
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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 substituting $t=\cos\frac{\theta}8$, but the professor…
Krang Lee
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Using calculus to find the winding number of $\sin(2t)+i\cos(t)$

To calculate an integral I need to find the winding number of $$\gamma(t)=\sin(2t)+i\cos(t)$$ around $\frac{i}{2}$. Graphically it looks like it is $1$. How can I use calculus to show this rigorously? I don't know how to calculate…
Ruben Kruepper
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