What is path dependent integral?
One obvious way to tell confirm that a vector field is path dependent is to compute a line integral of the vector field along multiple piecewise smooth curves connecting points P and Q. If the value of the line integral changes from one curve to the next, then the vector field is path dependent.
Is line integral path dependent?
Showing that if a vector field is the gradient of a scalar field, then its line integral is path independent.
How do you show a line integral is independent of the path?
Definitions
- The line integral of a vector function F = P i + Q j + R k is said to be path independent, if and only if and are continuous in a domain and if there exists some scalar function u = u ( x , y , z ) in such that.
- If this is the case, then the line integral of along the curve from to is given by the formula.
What is line integral formula?
Line Integral Formula r (a) and r(b) gives the endpoints of C and a < b. Line integral Formula for Vector Field. For a vector field with function, F: U ⊆ Rn → Rn, a line integral along with a smooth curve C ⊂ U, in the direction “r” is defined as: ∫C F(r).
What is path dependent function?
Path functions are functions that depend on the path taken to reach that specific value . Thus, a path function is a property or value that is dependent on the path taken to establish that value.
When an integral is path independent?
An integral is path independent if it only depends on the starting and finishing points. Consequently, on any curve C={r(t)|t∈[a,b]}, by the fundamental theorem of calculus ∫CFdr=∫C∇fdr=f(r(b))−f(r(a)), in other words the integral only depends on r(b) and r(a): it is path independent.
What does Green’s theorem calculate?
In summary, we can use Green’s Theorem to calculate line integrals of an arbitrary curve by closing it off with a curve C0 and subtracting off the line integral over this added segment. Another application of Green’s Theorem is that is gives us one way to calculate areas of regions.
What are path dependent variables?
Path Dependent vs Path Independent Work Path dependent variables are variables in which the path between two points that an object takes matters for that variable. Friction is an example of a force that is path dependent. In the presence of friction, some kinetic energy is always transformed into thermal energy.
What does path dependent mean in physics?
Path dependence implies that the amount of work or heat needed to make the change depends on how the process was performed, not just what state the material started in and ended in.
Is the line integral of a vector field path dependent?
The above result states that the line integral of a vector field derived from a gradient depends only on the function f(x,y,z) and on the initial point (x(a),y(a),z(a)) and final point (x(b),y(b),z(b)) and not on the particular curve C. Hence, the integral is path independent. (Compare this with an example of a path dependent line integral.)
Which is the best description of the path integral?
For integrals along a path, also known as line or contour integrals, see line integral. The path integral formulation is a description in quantum mechanics that generalizes the action principle of classical mechanics.
How is the path integral related to the Schrodinger equation?
The Schrödinger equation is a diffusion equation with an imaginary diffusion constant, and the path integral is an analytic continuation of a method for summing up all possible random walks.
How is path integral formulation different from other formulations?
Whereas in quantum mechanics the path integral formulation is fully equivalent to other formulations, it may be that it can be extended to quantum gravity, which would make it different from the Hilbert space model. Feynman had some success in this direction, and his work has been extended by Hawking and others.