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17. VECTOR CALCULUS. VECTOR CALCULUS. 17.8 Stokes’ Theorem. In this section, we will learn about: The Stokes’ Theorem and using it to evaluate integrals. . STOKES’ VS. GREEN’S THEOREM. Stokes’ Theorem can be regarded as a higher-dimensional version of Green’s Theorem.

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## VECTOR CALCULUS

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**17**VECTOR CALCULUS**VECTOR CALCULUS**17.8 Stokes’ Theorem • In this section, we will learn about: • The Stokes’ Theorem and • using it to evaluate integrals.**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**• The figure shows an oriented surface with unit normal vector n. • The orientation of Sinduces the positive orientation of the boundary curve C. Fig. 17.8.1, p. 1129**INTRODUCTION**• This means that: • If you walk in the positive direction around Cwith your head pointing in the direction of n, the surface will always be on your left. Fig. 17.8.1, p. 1129**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 that contains S. • Then,**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 Sof the tangential component of F is equal to the surface integral of the normal component of the curl of F.**STOKES’ THEOREM**Equation 1 • The positively oriented boundary curve of the oriented surface S is often written as ∂S. • So,the theorem can be expressed as:**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 boundaryof S.**STOKES’ THEOREM, GREEN’S THEOREM, & FTC**• In fact, consider the special case where the surface S: • Is flat. • Lies in the xy-plane with upward orientation.**STOKES’ THEOREM, GREEN’S THEOREM, & FTC**• Then, • The unit normal is k. • The surface integral becomes a double integral. • Stokes’ Theorem becomes:**STOKES’ THEOREM, GREEN’S THEOREM, & FTC**• This is precisely the vector form of Green’s Theorem given in Equation 12 in Section 17.5 • Thus, we see that Green’s Theorem is really a special case of Stokes’ Theorem.**STOKES’ THEOREM**• Stokes’ Theorem is too difficult for us to prove in its full generality. • Still, we can give a proof when: • S is a graph. • F, S, and C are well behaved.**STOKES’ TH.—SPECIAL CASE**Proof • We assume that the equation of Sis: z = g(x, y), (x, y) Dwhere: • g has continuous second-order partial derivatives. • D is a simple plane region whose boundary curve C1 corresponds to C.**STOKES’ TH.—SPECIAL CASE**Proof • If the orientation of S is upward, the positive orientation of C corresponds to the positive orientation of C1. Fig. 16.8.2, p. 1129**STOKES’ TH.—SPECIAL CASE**Proof • We are also given that: • F = P i + Q j + R kwhere the partial derivatives of P, Q, and R are continuous.**STOKES’ TH.—SPECIAL CASE**Proof • S is a graph of a function. • Thus, we can apply Formula 10 in Section 17.7 with F replaced by curl F.**STOKES’ TH.—SPECIAL CASE**Proof—Equation 2 • The result is: • where the partial derivatives of P, Q, and Rare evaluated at (x, y, g(x, y)).**STOKES’ TH.—SPECIAL CASE**Proof • Suppose • x =x(t) y =y(t) a ≤t ≤b • is a parametric representation of C1. • Then, a parametric representation of Cis: x =x(t) y =y(t) z =g(x(t), y(t)) a ≤t ≤b**STOKES’ TH.—SPECIAL CASE**Proof • This allows us, with the aid of the Chain Rule, to evaluate the line integral as follows:**STOKES’ TH.—SPECIAL CASE**Proof • We have used Green’s Theorem in the last step.**STOKES’ TH.—SPECIAL CASE**Proof • Next, we use the Chain Rule again, remembering that: • P, Q, and R are functions of x, y, and z. • z is itself a function of x and y.**STOKES’ TH.—SPECIAL CASE**Proof • Thus, we get:**STOKES’ TH.—SPECIAL CASE**Proof • Four terms in that double integral cancel. • The remaining six can be arranged to coincide with the right side of Equation 2. • Hence,**STOKES’ THEOREM**Example 1 • Evaluate where: • F(x, y, z) = –y2i + x j + z2k • 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.)**STOKES’ THEOREM**Example 1 • The curve C (an ellipse) is shown here. • could be evaluated directly. • However, it’s easier to use Stokes’ Theorem. Fig. 17.8.3, p. 1131**STOKES’ THEOREM**Example 1 • We first compute:**STOKES’ THEOREM**Example 1 • 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. Fig. 17.8.3, p. 1131**STOKES’ THEOREM**Example 1 • The projection D of S on the xy-plane is the disk x2 + y2≤ 1. • So, using Equation 10 in Section 17.7 with z =g(x, y) = 2 – y, we have the following result. Fig. 17.8.3, p. 1131**STOKES’ THEOREM**Example 1**STOKES’ THEOREM**Example 2 • 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. Fig. 17.8.4, p. 1131**STOKES’ THEOREM**Example 2 • To find the boundary curve C, we solve: x2 + y2 + z2 = 4 and x2 + y2 = 1 • Subtracting, we get z2 = 3. • So, (since z > 0). Fig. 17.8.4, p. 1131**STOKES’ THEOREM**Example 2 • So, C is the circle given by: x2 + y2 = 1, Fig. 17.8.4, p. 1131**STOKES’ THEOREM**Example 2 • 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:**STOKES’ THEOREM**Example 2 • Thus, by Stokes’ Theorem,**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!**STOKES’ THEOREM**Equation 3 • In general, if S1 and S2 are oriented surfaces with the same oriented boundary curve Cand 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.**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.**CURL VECTOR**• Consider the line integral and recall that v ∙T is the component of vin 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.**CIRCULATION**• Thus, is a measure of the tendency of the fluid to move around C. • It iscalled the circulation of v around C. Fig. 17.8.5, p. 1132**CURL VECTOR**• Now, let: P0(x0, y0, z0) be a point in the fluid. • 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.**CURL VECTOR**• Thus, by Stokes’ Theorem, we get the following approximation to the circulation around the boundary circle Ca:**CURL VECTOR**Equation 4 • The approximation becomes better as a→ 0. • Thus, we have:**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.**CURL & CIRCULATION**• 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. Fig. 17.8.6, p. 1132**CLOSED CURVES**• Finally, we mention that Stokes’ Theorem can be used to prove Theorem 4 in Section 16.5: • If curl F = 0 on all of , then F is conservative.**CLOSED CURVES**• From Theorems 3 and 4 in Section 17.3, we know that F is conservative if for every closed path C. • Given C, suppose we can find an orientable surface S whose boundary is C. • This can be done, but the proof requires advanced techniques.**CLOSED CURVES**• Then, Stokes’ Theorem gives: • A curve that is not simple can be broken into a number of simple curves. • The integrals around these curves are all 0.**CLOSED CURVES**• Adding these integrals, we obtain: for any closed curve C.

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