Please write a report about Stokes' theorem and its applications with examples.
here are some links as references:
https://tutorial.math.lamar.edu/classes/calcIII/stokestheorem.aspx
https://en.wikipedia.org/wiki/Stokes%27_theorem
https://en.wikipedia.org/wiki/Curl_(mathematics)
Slide 1 16.8 Stokes’ Theorem In this section, we will learn about: The Stokes’ Theorem and using it to evaluate integrals. VECTOR CALCULUS STOKES’ VS. GREEN’S THEOREM Stokes’ Theorem can be regarded as a higher-dimensional version of Green’s Theorem. Green’s Theorem relates a double integral over a plane region D to a line integral around its plane boundary curve. Stokes’ Theorem relates a surface integral over a surface S to a line integral around the boundary curve of S (a space curve). INTRODUCTION Oriented surface with unit normal vector n. The orientation of S induces the positive orientation of the boundary curve C. 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. STOKES’ THEOREM Let: S be an oriented piecewise-smooth surface bounded by a simple, closed, piecewise-smooth boundary curve C with positive orientation. F be a vector field whose components have continuous partial derivatives on an open region in R3 that contains S. Then, curl C S d d⋅ = ⋅∫ ∫∫F r F S STOKES’ THEOREM The theorem is named after the Irish mathematical physicist Sir George Stokes (1819–1903). What we call Stokes’ Theorem was actually discovered by the Scottish physicist Sir William Thomson (1824–1907, known as Lord Kelvin). Stokes learned of it in a letter from Thomson in 1850. STOKES’ THEOREM Thus, Stokes’ Theorem says: The line integral around the boundary curve of S of the tangential component of F is equal to the surface integral of the normal component of the curl of F. and curl curl C C S S d ds d d ⋅ = ⋅ ⋅ = ⋅ ∫ ∫ ∫∫ ∫∫ F r F T F S F n S STOKES’ THEOREM The positively oriented boundary curve of the oriented surface S is often written as ∂S. So, the theorem can be expressed as: curl S S d d ∂ ⋅ = ⋅∫∫ ∫F S F r Equation 1 STOKES’ THEOREM, GREEN’S THEOREM, & FTC There is an analogy among Stokes’ Theorem, Green’s Theorem, and the Fundamental Theorem of Calculus (FTC). As before, there is an integral involving derivatives on the left side of Equation 1 (recall that curl F is a sort of derivative of F). The right side involves the values of F only on the boundary of S. STOKES’ THEOREM, GREEN’S THEOREM, & FTC In fact, consider the special case where the surface S is flat, in the xy-plane with upward orientation. Then: The unit normal is k. The surface integral becomes a double integral. Stokes’ Theorem becomes: Thus, we see that Green’s Theorem is really a special case of Stokes’ Theorem. ( )curl curl C S S d d dA⋅ = ⋅ = ⋅∫ ∫∫ ∫∫F r F S F k STOKES’ THEOREM Evaluate , where: F(x, y, z) = –y2 i + x j + z2 k C is the curve of intersection of the plane y + z = 2 and the cylinder x2 + y2 = 1. (Orient C to be counterclockwise when viewed from above.) could be evaluated directly, however, it’s easier to use Stokes’ Theorem. C d⋅∫ F r Example 1 C d⋅∫ F r STOKES’ THEOREM We first compute for F(x, y, z) = –y2 i + x j + z2 k: ( ) 2 2 curl 1 2y x y z y x z ∂ ∂ ∂ = = + ∂ ∂ ∂ − i j k F k Example 1 STOKES’ THEOREM There are many surfaces with boundary C. The most convenient choice, though, is the elliptical region S in the plane y + z = 2 that is bounded by C. If we orient S upward, C has the induced positive orientation. Example 1 STOKES’ THEOREM The projection D of S on the xy-plane is the disk x2 + y2 ≤ 1. So, using Equation 10 in Section 16.7 with z = g(x, y) = 2 – y, we have the following result. Example 1 STOKES’ THEOREM ( ) ( ) ( ) ( ) 2 1 0 0 12 32 0 0 2 1 2 2 30 1 2 curl 1 2 1 2 sin 2 sin 2 3 sin 2 0 C S D d d y dA r r dr d r r d d π π π θ θ θ θ θ θ π π ⋅ = ⋅ = + = + = + = + = + = ∫ ∫∫ ∫∫ ∫ ∫ ∫ ∫ F r F S Example 1 STOKES’ THEOREM Use Stokes’ Theorem to compute where: F(x, y, z) = xz i + yz j + xy k S is the part of the sphere x2 + y2 + z2 = 4 that lies inside the cylinder x2 + y2 =1 and above the xy-plane. curl S d⋅∫∫ F S Example 2 STOKES’ THEOREM To find the boundary curve C, we solve: x2 + y2 + z2 = 4 and x2 + y2 = 1 Subtracting, we get z2 = 3, and (since z > 0), So, C is the circle given by: x2 + y2 = 1, Example 2 3z = 3z = STOKES’ THEOREM A vector equation of C is: r(t) = cos t i + sin t j + k 0 ≤ t ≤ 2π Therefore, r’(t) = –sin t i + cos t j Also, we have: 3 Example 2 ( )( ) 3 cos 3 sin cos sint t t t t= + +F r i j k STOKES’ THEOREM Thus, by Stokes’ Theorem, ( ) 2 0 2 0 2 0 curl ( ( )) '( ) 3 cos sin 3 sin cos 3 0 0 C S d d t t dt t t t t dt dt π π π ⋅ = ⋅ = ⋅ = − + = = ∫∫ ∫ ∫ ∫ ∫ F S F r F r r Example 2 STOKES’ THEOREM •Note that, in Example 2, we computed a surface integral simply by knowing the values of F on the boundary curve C. •This means that: If we have another oriented surface with the same boundary curve C, we get exactly the same value for the surface integral! •In general, if S1 and S2 are oriented surfaces with the same oriented boundary curve C and both satisfy the hypotheses of Stokes’ Theorem, then This fact is useful when it is difficult to integrate over one surface but easy to integrate over the other. 1 2 curl curl C S S d d d⋅ = ⋅ = ⋅∫∫ ∫ ∫∫F S F r F S CURL VECTOR We now use Stokes’ Theorem to throw some light on the meaning of the curl vector. Suppose that C is an oriented closed curve and v represents the velocity field in fluid flow. Consider the line integral and recall that v ∙ T is the component of v in the direction of the unit tangent vector T. This means that the closer the direction of v is to the direction of T, the larger the value of v ∙ T. C C d ds⋅ = ⋅∫ ∫v r v T CIRCULATION Thus, is a measure of the tendency of the fluid to move around C. It is called the circulation of v around C. C d⋅∫ v r CURL VECTOR •Now, let: P0(x0, y0, z0) be a point in the fluid, and Sa be a small disk with radius a and center P0. Then, (curl F)(P) ≈ (curl F)(P0) for all points P on Sa because curl F is continuous. Thus, by Stokes’ Thm., we get the following approximation to the circulation around the boundary circle Ca: • The approximation becomes better as a → 0. Thus, we have: ( ) ( ) ( ) ( ) 0 0 2 0 0 curl curl curl curl a a a a C S S S d d dS P P dS P P aπ ⋅ = ⋅ = ⋅ ≈ ⋅ = ⋅ ∫ ∫∫ ∫∫ ∫∫ v r v S v n v n v n ( ) ( )0 0 20 1curl lim aCa P P d aπ→ ⋅ = ⋅∫v n v r CURL & CIRCULATION Equation 4 gives the relationship between the curl and the circulation. It shows that curl v ∙ n is a measure of the rotating effect of the fluid about the axis n. The curling effect is greatest about the axis parallel to curl v. Imagine a tiny paddle wheel placed in the fluid at a point P. The paddle wheel rotates fastest when its axis is parallel to curl v. 16.9 The Divergence Theorem In this section, we will learn about: The Divergence Theorem for simple solid regions, and its applications in electric fields and fluid flow. VECTOR CALCULUS INTRODUCTION In Section 16.5, we rewrote