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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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 $\int_C (z^2,xz,2xy)\cdot dr$ 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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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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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 ...