Stokes theorem is used in differential geometry. Stokes Theorem describes about how the differential forms on the manifolds are integrated.
Stokes Theorem Definition
If we integrate a differential form over a boundary that is of some oriental manifold, then it is equal to the value obtained by integrating its exterior derivative of the differential form over the whole manifold.
Symbolically, Stokes theorem is represented by:
∫C F.dr = ∫∫ curl F.ds
∫∫ curl F.ds = ∫∫ curl F.n ds
Stokes theorem can be defined as a theorem that will relate the surface interval to that of the line interval. The surface integral of normal component of curl F is present above the surface called S. The line integral is defined in the region of the space curve C of the surface S of the tangential component of F.
As per the fundamental theorem of calculus, integration a function g which is over the interval [m, n] is computed by finding the anti derivative G of the function g. Stokes Theorem can be considered as a vast generalization of the fundamental theorem of calculus.
Know more about Stokes Theorem. Between, if you have problem in learning and solving Calculus problems please browse expert math related websites like mathcaptain.com,tutorvista.com for more help.
Stokes Theorem Example
Let us consider an example of integrating G over the manifold C. Given G(a,b,c) equal to –b ^2 x +ay +c^2z. The curve C is obtained by the intersection of the plane and cylinder. The equation of the plane is given by the sum of b and c, which is equal to 2. The equation of the cylinder is given by the summation of square of “a” and square of “b” which is equal to 1. When viewed from the upper side, the curve C should have been oriented counterclockwise.
We can use Stokes Theorem to evaluate the above example. The first thing we should do is to compute curl G using the coefficients of x, y and z i.e. “b^2”, “a” and “c^2” respectively. We get the value of curl G as (1+2b) multiplied with z.
The elliptical region in the plane b+c=2 is chosen. The surface S is oriented upward and so the boundary curve C is induced with positive orientation. The projection of the surface of S is given by the equation summation of the square of “a” and the square of “b” less than or equal to 1. By applying the equation c = g(a,b) = 2-b, we get half multiplied with 22/7 or pie.
The above example is an example of single integral. Stokes theorem can also be used to evaluate double integral. We can Verify Stokes Theorem by Stokes Theorem Proof.
Stokes Theorem Definition
If we integrate a differential form over a boundary that is of some oriental manifold, then it is equal to the value obtained by integrating its exterior derivative of the differential form over the whole manifold.
Symbolically, Stokes theorem is represented by:
∫C F.dr = ∫∫ curl F.ds
∫∫ curl F.ds = ∫∫ curl F.n ds
Stokes theorem can be defined as a theorem that will relate the surface interval to that of the line interval. The surface integral of normal component of curl F is present above the surface called S. The line integral is defined in the region of the space curve C of the surface S of the tangential component of F.
As per the fundamental theorem of calculus, integration a function g which is over the interval [m, n] is computed by finding the anti derivative G of the function g. Stokes Theorem can be considered as a vast generalization of the fundamental theorem of calculus.
Know more about Stokes Theorem. Between, if you have problem in learning and solving Calculus problems please browse expert math related websites like mathcaptain.com,tutorvista.com for more help.
Stokes Theorem Example
Let us consider an example of integrating G over the manifold C. Given G(a,b,c) equal to –b ^2 x +ay +c^2z. The curve C is obtained by the intersection of the plane and cylinder. The equation of the plane is given by the sum of b and c, which is equal to 2. The equation of the cylinder is given by the summation of square of “a” and square of “b” which is equal to 1. When viewed from the upper side, the curve C should have been oriented counterclockwise.
We can use Stokes Theorem to evaluate the above example. The first thing we should do is to compute curl G using the coefficients of x, y and z i.e. “b^2”, “a” and “c^2” respectively. We get the value of curl G as (1+2b) multiplied with z.
The elliptical region in the plane b+c=2 is chosen. The surface S is oriented upward and so the boundary curve C is induced with positive orientation. The projection of the surface of S is given by the equation summation of the square of “a” and the square of “b” less than or equal to 1. By applying the equation c = g(a,b) = 2-b, we get half multiplied with 22/7 or pie.
The above example is an example of single integral. Stokes theorem can also be used to evaluate double integral. We can Verify Stokes Theorem by Stokes Theorem Proof.
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