VECTOR CALCULUS

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16. VECTOR CALCULUS. VECTOR CALCULUS. 16.8 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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16

VECTOR CALCULUS

VECTOR CALCULUS

16.8

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.
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.
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 16.5
• 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.
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 16.7 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.
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.
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 16.7 with z =g(x, y) = 2 – y, we have the following result.
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.
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).
STOKES’ THEOREM

Example 2

• So, C is the circle given by: x2 + y2 = 1,
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.
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.
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 16.3, 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.