Math 2374: Multivariable Calculus and Vector Analysis

Transcription

1 Math 2374: Multivariable Calulus and Vetor Analysis Part 26 Fall 2012

2 The integrals of multivariable alulus line integral of salar-valued funtion line integral of vetor fields surfae integral of salar-valued funtion surfae integral of vetor fields double integral triple integral

3 Line integral of salar-valued funtion Definition (Path integral) Let f : U R 3 R be a ontinuous funtion. Let : I = [a, b] U of lass C 1. We suppose that the omposition funtion t f (x(t), y(t), z(t)) is ontinuous on I. We define the path integral as: Remark f ds = b a f (x(t), y(t), z(t)) (t) dt When f = 1, we reover the definition of the ar length of.

4 Line integral of vetor fields Definition (Line integral) Let F : U R 3 R 3 be a ontinuous vetor fields. Let : I = [a, b] U of lass C 1. We suppose that the omposition funtion t F(x(t), y(t), z(t)) is ontinuous on I. We define the line integral as: F ds = b a F(x(t), y(t), z(t)) (t)dt

5 How to ompute line integrals? 1 Diretly by applying the formula given in the definition. 2 If F is onservative (F = f ) then we have: F ds = f ((b)) f ((a)). 3 If F and are in R 2 and is a simple lose urve: ( F2 F ds = x F ) 1 dxdy (Green s theorem) y D ( is the boundary of D and is oriented ounterlokwise). 4 If F and are in R 3 and is a simple lose urve: F ds = urlf ds (Stokes theorem) S ( must be a positively oriented boundary of S)

6 Parametrized Surfaes Definition (Parametrized Surfaes) A parametrized surfae is a funtion Φ : D R 2 R 3, where D is a domain in R 2. The surfae S orresponding to the funtion Φ is its image: S = Φ(D). We an write: Φ(u, v) = (x(u, v), y(u, v), z(u, v)). When Φ is of lass C 1 we define the two tangent vetors: T u = Φ u T v = Φ v = x u i + y u j + z u k = x v i + y v j + z v k S is regular at Φ(u 0, v 0 ) provided that T u T v 0 at (u 0, v 0 ).

7 Surfae integral of salar-valued funtion Definition If f (x, y, z) is a real-valued ontinuous funtion defined on the parametrized surfae S, we define the integral of f over S to be f ds = f (Φ(u, v)) T u T v dudv Remark S D If f = 1 then we reover the definition of the area of the surfae S: A(s) = T u T v dudv D

8 Surfae integral of vetor fields Definition Let F be a vetor field defined on the parametrized surfae S, we define the surfae integral of F over S to be F ds = F(Φ(u, v)) (T u T v ) dudv S D Remark 1 If F = url G and is the positively oriented boundary of S F ds = G ds (Stokes theorem) S 2 If S is a losed surfae that is the boundary of a solid W F ds = div FdV (Gauss theorem) S W

9 Change of variables in double integral Definition Let D and D be two elementary regions in the plane and let T : D D be of lass C 1 ; suppose that T is one-to-one on D and T (D ) = D. Then for any integrable funtion f : D R, we have f (x, y)dxdy = f (x(u, v), y(u, v)) (x, y) D D (u, v) dudv Polar oordinates f (x, y)dxdy = f (r os θ, r sin θ)rdrdθ D D

10 Change of variables in triple integral Definition Let W and W be two elementary regions in spae and let T : W W be of lass C 1 ; suppose that T is one-to-one on w and T (W ) = W. Then for any integrable funtion f : W R, we have f (x(u, v), y(u, v), z(u, v)) (x, y, z) W (u, v, w) dudvdw Cylindrial oordinates Spherial oordinates W f (r os θ, r sin θ, z)rdrdθdz W f (ρ sin φ os θ, ρ sin φ sin θ, ρ os φ)ρ 2 sin φdρdθdφ

11 The fundamental theorems of vetor alulus the gradient theorem for line integrals Green s theorem Stokes theorem Gauss theorem (divergene theorem)

12 Gradient theorem for line integrals Theorem Let f : U R 3 R be a funtion of lass C 1. Let : I = [a, b] U of lass C 1. We suppose that the omposition funtion t f ((t)) is ontinuous on I. We have f ds = f ((b)) f ((a)).

13 Green s theorem Theorem Let D be a simple region and let be its boundary. Suppose that P : D R and Q : D R are of lass C 1. Then ( Q Pdx + Qdy = + D x P ) dxdy y Remark Can be generalized to any deent region in R 2. If is a simple losed urve that bounds a region to whih Green s theorem applies, then the area of the region D bounded by = D is A = 1 xdy ydx. 2

14 Stokes theorem Theorem Let S be an oriented surfae defined by a one-to-one parametrization Φ : D R 2 S, where D is a region to whih Green s theorem applies. Let S denote the oriented boundary of S and let F be a C 1 vetor field on S. Then ( F) ds = F ds. Remark S If S has no boundary then the integral on the left is zero. S

15 Gauss theorem Theorem Let W be a solid and S its boundary: S = W. Let F be a C 1 vetor field defined on W then ( F)dV = F ds. W S