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Meaning of line- path- and contour integrals. A curve-, line-, path- or contour integral extends the usual definition of an integral to the integration in the complex plane or in a multidimensional space. The term contour integral is typically reserved for line integrals in the complex plane but does not imply integration over a closed contour.
Path integral of the electric field from $(1,0)$ to $(0,1)$ 1. Proof that a central field is conservative ...
The path integral in momentum representation is a mathematical tool used in quantum mechanics to calculate the probability of a particle moving from one point to another in a given time interval. It involves integrating over all possible paths that the particle can take in momentum space.
But we want to do it the hard way: by a path integral. So we take: $\mathbf{r}(t) = 2x^{n} - 1$ (this just goes from -1 to 1 as t goes from 0 to 1) and solve the following integral instead: $\int_{0}^1(2x^{n} - 1)^2\>dt$ This is your (2). Now note what happens for big k (this is the graph for $\mathbf{r}(t) = 2x^{101} - 1$):
First note that Polar Coordinates are orthogonal coordinates, so the unit vectors $\mathbf{\hat r}$ and $\mathbf{\hat \theta}$ are orthogonal.
Path integral of ${1\over z^{2}}$ around a circle ... gamma:[a,b]\to \Bbb C$, and since ${1\over z}$ is a ...
Inside the path integral, imagine that we're summing an action of classical fields over a measure of all possible values for those classical fields. As a variable of integration, $\phi$ is thus a classical field and each value $\phi(x_i)$ is a scalar value. Outside the path integral is where $\phi$ is a Hilbert space operator.
path integral definition of determinant. Ask Question Asked 8 years, 3 months ago. Modified 8 years, 3 ...
For a conservative field $\vec{F}$ the curve integral $\oint_\gamma \vec{F} \cdot d\vec{x} = 0$ for every closed path $\gamma$. If a field is not conservative, the integral can still be zero for a lot of closed curves, but not for every curve.
Path integral quantization is a mathematical framework used in quantum field theory to describe the dynamics of particles and fields. It involves integrating over all possible paths that a particle or field can take, taking into account their interactions with other particles and fields.