Examples of Stokes' theorem in the following topics:
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- Stokes' theorem relates the integral of the curl of a vector field over a surface to the line integral of the field around the boundary.
- The generalized Stokes' theorem is a statement about the integration of differential forms on manifolds, which both simplifies and generalizes several theorems from vector calculus.
- The Kelvin–Stokes theorem, also known as the curl theorem, is a theorem in vector calculus on $R^3$.
- The Kelvin–Stokes theorem is a special case of the "generalized Stokes' theorem."
- An illustration of the Kelvin–Stokes theorem, with surface $\Sigma$, its boundary $\partial$, and the "normal" vector $\mathbf{n}$.
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- Surfaces that occur in two of the main theorems of vector calculus, Stokes' theorem and the divergence theorem, are frequently given in a parametric form.
- We will study surface integral of vector fields and related theorems in the following atoms.
- An illustration of the Kelvin–Stokes theorem, with surface $\Sigma$, its boundary $\partial$, and the "normal" vector $\mathbf{n}$.
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- Green's theorem gives relationship between a line integral around closed curve $C$ and a double integral over plane region $D$ bounded by $C$.
- Green's theorem is a special case of the Kelvin–Stokes theorem, when applied to a region in the $xy$-plane.
- Considering only two-dimensional vector fields, Green's theorem is equivalent to the two-dimensional version of the divergence theorem.
- Green's theorem can be used to compute area by line integral.
- Explain the relationship between the Green's theorem, the Kelvin–Stokes theorem, and the divergence theorem
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- We have also studied theorems linking derivatives and integrals of single variable functions.
- The theorems we learned are gradient theorem, Stokes' theorem, divergence theorem, and Green's theorem.
- In a more advanced study of multivariable calculus, it is seen that these four theorems are specific incarnations of a more general theorem, the generalized Stokes' theorem, which applies to the integration of differential forms over manifolds.
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- An illustration of the Kelvin–Stokes theorem, with surface $\Sigma$, its boundary $\partial$, and the "normal" vector $n$.
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- The divergence theorem, also known as Gauss's theorem or Ostrogradsky's theorem, relates the flow (that is, flux) of a vector field through a surface to the behavior of the vector field inside the surface.
- In physics and engineering, the divergence theorem is usually applied in three dimensions.
- In one dimension, it is equivalent to the fundamental theorem of calculus.
- The theorem is a special case of the generalized Stokes' theorem.
- Apply the divergence theorem to evaluate the outward flux of a vector field through a closed surface
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- Using the Stokes' theorem in vector calculus, the left hand side is$\oint_C \vec E \cdot d\vec s = \int_S (\nabla \times \vec E) \cdot d\vec A$.
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- The corresponding form of the fundamental theorem of calculus is Stokes' theorem, which relates the surface integral of the curl of a vector field to the line integral of the vector field around the boundary curve.
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