1 Chapter 10 Vector Integral Calculus Integral Theorems

1 Chapter 10. Vector Integral Calculus. Integral Theorems EMLAB

1. 1 Line Integrals 2 Definition and Evaluation of Line Integrals For closed path : EMLAB

3 EXAMPLE 1 Evaluation of a Line Integral in the Plane Solution. EMLAB

4 THEOREM 1 Direction-Preserving Parametric Transformations Any representations of C that give the same positive direction on C also yield the same value of the line integral PROOF EMLAB

5 Motivation of the Line Integral : Work Done by a Force The sum of these n works is Other Forms of Line Integrals EMLAB

6 EXAMPLE 5 EMLAB

7 THEOREM 2 Path Dependence The line integral generally depends not only on F and on the endpoints A and B of the path, but also on the path itself along which the integral is taken. EMLAB

10. 2 Path Independence of Line Integrals 8 Path Independence : A line integral takes on the same value no matter what path of integration is taken. Path independence in a domain D holds if and only if: EMLAB

9 Conditions for path independence EMLAB

10 Test for exactness EMLAB

11 EXAMPLE 2 Path Independence. Determination of a Potential Evaluate the integral from A(0, 1, 2) to B(1, -1, 7) by showing that F has a potential. Solution. If F has a potential f, we should have EMLAB

EXAMPLE 3 Exactness and Independence of Path. Determination of a Potential 12 Show that the differential form under the integral sign of EMLAB

13 EXAMPLE 4 On the Assumption of Simple Connectedness in the proof EMLAB

14 EMLAB

10. 3 Calculus Review: Double Integrals. 15 Fig. 227. Subdivision of a region R EMLAB

Evaluation of Double Integrals by Two Successive Integrations 16 Fig. 229. Evaluation of a double integral EMLAB

Applications of Double Integrals 17 Area A of a region R in the xy-plane is Then the total mass M in R is the center of gravity of the mass in R EMLAB

18 the moments of inertia the polar moment of inertia EMLAB

Change of Variables in Double Integrals. Jacobian 19 change of the variables of integration in integrals. The formula for a change of variables in double integrals from x, y to u, v is EMLAB

EXAMPLE 1 Change of Variables in a Double Integral 20 Evaluate the following double integral over the square R in Fig. 232. Region R in Example 1 EMLAB

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EXAMPLE 2 Double Integrals in Polar Coordinates. Center of Gravity. Moments of Inertia 22 The center of gravity has the coordinates For reasons of symmetry. EMLAB

23 The moments of inertia are For reasons of symmetry. EMLAB

10. 4 Green’s Theorem in the Plane 24 Green’s Theorem in the Plane (Transformation between Double Integrals and Line Integrals) Here we integrate along the entire boundary C of R in such a sense that R is on the left as we advance in the direction of integration EMLAB

25 PROOF We prove Green’s theorem in the plane, first for a special region R that can be represented in both forms Fig. 235. Example of a special region EMLAB

26 EMLAB

27 Fig. 237. Proof of Green’s theorem Some Applications of Green’s Theorem EXAMPLE 2 Area of a Plane Region as a Line Integral Over the Boundary EMLAB

28 For an ellipse EXAMPLE 3 Area of a Plane Region in Polar Coordinates EMLAB

29 As an application, we consider the cardioid Fig. 239. Cardioid EMLAB

EXAMPLE 4 Transformation of a Double Integral of the Laplacian of a Function into a Line Integral of Its Normal Derivative 30 EMLAB

Example 1 31 EMLAB

32 Example 2 EMLAB

10. 6 Surface Integrals 33 Representation of Surfaces Example For surfaces S in surface integrals, it will often be more practical to use a parametric representation. Surfaces are two-dimensional. Hence we need two parameters. Fig. 241. Parametric representations of a curve and a surface EMLAB

EXAMPLE 1 Parametric Representation of a Cylinder 34 Fig. 242. Parametric representation of a cylinder EMLAB

EXAMPLE 2 Parametric Representation of a Sphere 35 Fig. 243. Parametric representation of a cylinder of a sphere EXAMPLE 3 Parametric Representation of a Cone EMLAB

Tangent Plane and Surface Normal 36 Fig. 244. Tangent plane and normal vector EXAMPLE 4 Unit Normal Vector of a Sphere EMLAB

37 EXAMPLE 5 Unit Normal Vector of a cone EMLAB

10. 6 Surface Integrals 38 ; unit normal vector EMLAB

39 EXAMPLE 1 Flux Through a Surface Compute the flux of water through the parabolic cylinder Solution. Fig. 245. Surface S in Example 1 EMLAB

40 EXAMPLE 2 Surface Integral Solution. Fig. 246. Portion of a plane in Example 2 EMLAB

Surface Integrals Without Regard to Orientation 41 mean value theorem for surface integrals If R in the above integral is simply connected and G(r) is continuous in a domain containing R, then there is a point in R such that If G = 1, (6) gives the area A(S) of S, EMLAB

42 EXAMPLE 4 Area of a Sphere EMLAB

43 EXAMPLE 5 Torus Surface (Doughnut Surface): Representation and Area Fig. 249. Torus in Example 5 EMLAB

44 EMLAB

10. 7 Triple Integrals. Divergence Theorem of Gauss 45 EMLAB

Divergence Theorem of Gauss 46 (Transformation Between Triple and Surface Integrals) EMLAB

47 Proof : Fig. 253. Example of a special region EMLAB

EXAMPLE 1 Evaluation of a Surface Integral by the Divergence Theorem 48 Evaluate where S is the closed surface in Fig. 252 consisting of the cylinder Fig. 252. Surface S Solution. EMLAB

EXAMPLE 2 Verification of the Divergence Theorem 49 Evaluate Solution. (a) Volume integral (a) Surface integral EMLAB

50 EMLAB

Mean value theorem for triple integrals 51 EMLAB

10. 8 Further Applications of the Divergence Theorem 52 EXAMPLE 4 Green’s Theorems Green’s first formula Green’s second formula EMLAB

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10. 9 Stokes’ Theorem 54 Fig. 254. Stokes’ theorem EMLAB

55 Fig. 256. Proof of Stokes’s theorem EMLAB

56 EMLAB

57 EXAMPLE 1 Verification of Stokes’s Theorem Evaluate a line integral along the perimeter of the paraboloid. Fig. 255. Surface S in Example 1 EMLAB