State and prove stock's theorm
Answers
Answer:
We will prove Stokes' theorem for a vector field of the form P (x, y, z) k . That is, we will show, with the usual notations, (3) P (x, y, z) dz = curl (P k ) · n dS . We assume S is given as the graph of z = f(x, y) over a region R of the xy-plane; we let C be the boundary of S, and C the boundary of R.
Explanation:
Where,
C = A closed curve.
S = Any surface bounded by C.
F = A vector field whose components have continuous derivatives in an open region of R3 containing S.
This classical declaration, along with the classical divergence theorem, fundamental theorem of calculus, and Green’s theorem are exceptional cases of the general formulation specified above. This means that:
If you walk in the positive direction around C with your head pointing in the direction of n, the surface will always be on your left.
S is an oriented smooth surface bounded by a simple, closed smooth-boundary curve C with positive orientation.
The Stoke’s theorem states that “the surface integral of the curl of a function over a surface bounded by a closed surface is equal to the line integral of the particular vector function around that surface.”
I hope this helps.