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9.2. Line Integrals Independent of Path Theorem 1 : Independence of path

9.2. Line Integrals Independent of Path Theorem 1 : Independence of path Above line integral with continuous F 1 , F 2 , F 3 in D in space is independent of path in D iff F=[F 1 , F 2 , F 3 ] is the gradient of some function f ( potential ) in D .

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9.2. Line Integrals Independent of Path Theorem 1 : Independence of path

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  1. 9.2. Line Integrals Independent of Path Theorem 1: Independence of path Above line integral with continuous F1, F2, F3 in D in space is independent of path in D iff F=[F1, F2, F3] is the gradient of some function f (potential) in D. Ex. 1) A(0,0,0)  B(2,2,2)

  2. C1 B C2 A • Ex. 2) • Integration Around Closed Curves and Independence of Path • Theorem 2: The integral is independent of path in a domain D iff its value around • every closed path in D is zero. • Exactness and Independence of Path • Path independence ~ gradient • integration around closed curves • exactness of the differential form • : exact in D if or • - Above form is exact iff there is a differentiable function f(x,y,z) in D such that

  3. A line integral is independent of path in D iff the differential form, has continuous F1, F2, F3 and is exact in D. • Theorem 3: • (a) If line integral is independent of path in D and thus the differential form • is exact in D, • (b) Above relation holds in D and D is simply connected, then line integral is indep. of • path in D. • A domain is simply connected if every closed curve in D can be continuously shrunk • to any point in D without leaving D. • Ex) simply connected: interior of a sphere or a cube, domain btw two concentric spheres… • not simply connected: a torus… • Ex. 3) • Ex. 4) On the assumption of simple connectedness (see textbook)

  4. h(x) y R d g(x) p(y) R b a x q(y) c x 9.3. Double Integrals • Double Integrals of f(x,y) over the region R • Some properties of double integrals • Mean value theorem: • Evaluation of Double Integrals y

  5. Change of Variables in Double Integrals. Jacobian • Ex) x=rcos, y=rsin • J=r

  6. 적분의 수치해석 (simple case 참고) Most common numerical integration schemes - replacing a complicated function or tabulated data with an easy integrable approx. func. (fn(x)=a polynomial func., ) Straight line fn. Parabola fn. • Trapezoidal Rule , first-order polynomial: By extending above formulation,

  7. Multiple Integrals: Simple case: use n equally spaced subintervals in x Trapezoidal rule:

  8. y R C1 C2 C** y x v(x) R C* u(x) b a x 9.4. Green Theorem in the Plane • Double integrals over a plane region  Line integralsover the boundary of the region • (easy evaluation of integrals) • Theorem 1: Green’s theorem in the plane (transformation btw double and line integrals) • R: a closed bounded region in xy plane • Boundary C • integrated along the entire boundary C of R such that R is on the • left in the direction of integration • Ex. 1)

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