[Anton, pp , pp ] & [Bourne, pp ]

Transcription

1 hapter 3 Integral Theorems [Anton, pp , pp ] & [Bourne, pp ] First of all some definitions which we will need in the following: Definition 3.1. (a) A domain (region) is an open connected subset of R n. (b) A domain R 3 is bounded, if there exists an R > 0 such that B R, where B R is the ball with radius R and centre 0. (c) A surface R 3 is open, if for all x 1, x 2 there exists a continuous curve from x 1 to x 2 which does not cross. A surface R 3 is closed, if it is not open. (d) A closed surface R 3 is convex, if every straight line intersects (meets) at two points at most. Examples. (e) A closed surface R 3 is semi convex, if we can choose a coordinate system 0xyz so that every straight line parallel to the coordinate axes intersects at two points at most. Examples. Note. Recall also (Remark 1.24) that a surface is smooth, if its parametrisation is continuously differentiable. is piecewise smooth, if = n i=1 i and i smooth. 30

2 3.1 The Divergence Theorem of Gauss Theorem 3.2 (Divergence Theorem). Let R 3 be a bounded domain with piecewise smooth, closed boundary (surface). uppose also that F : R 3 is a continuously differentiable vector field. Then F dv = F d. (3.1) Proof. (Only for smooth and semi convex). Let D be the projection of onto the (x, y) plane. onsider the line L through Pfrag the point replacements (x, y, 0) parallel to the z axis. ince is semi-convex, L intersects at two points (x, y, f(x, y)) T and (x, y, g(x, y)) T, where f(x, y) g(x, y) for all (x, y) D (otherwise change the coordinate system). z L (x, y, g(x, y)) 1 Hence, (x, y, f(x, y)) 0 y x (x, y, 0) D (i) Let us first show that F 3 k d = D { } F 3 (x, y, g(x, y)) F 3 (x, y, f(x, y)) dx dy. (3.2) 31

3 (ii) Now we show that F 3 z dv = F 3 k d. (3.3) (iii) imilarly, by projecting onto the (x, z)-plane and onto the (y, z)-plane we can establish F 2 y dv = F 2 j d, (3.4) F 1 x dv = F 1 i d, (3.5) and Remark 3.3. This proof can be extended in a straightforward way to domains with piecewise smooth and non-semi-convex boundary, if = n i=1 i, where each of the i has a smooth, semi-convex boundary i, e.g. torus. Example 3.4. Find F d where is the surface of the unit cube and F := (x2, y 2, z 2 ) T. orollary 3.5. Let and be as in Theorem 3.2. uppose f : R and F : R 3 are continuously differentiable. Then f dv = f d (3.6) F dv = F d (3.7) 32

4 Proof. Let a R 3 be constant. (i) Apply the Divergence Theorem to G := f a: (ii) Apply the Divergence Theorem to G := a F : 3.2 Green s Theorem in the Plane Note. In this section we work in R 2 not in R 3! Definition 3.6. (a) A closed curve R 2, is simple, if it does not intersect itself, e.g. Pfrag replacements simple not simple. (b) A closed curve R 2 is convex, if every straight line intersects at 2 points at most. (c) A closed curve R 2 is semi convex, if we can choose a coordinate system 0xy so that every straight line parallel to the coordinate axes intersects at 2 points at most. 33

5 Theorem 3.7 (Green s Theorem in the Plane). Let R 2 be a bounded domain with simple, piecewise smooth boundary (curve) R 2 described in the anticlockwise sense. uppose that Φ : R 2 is a continuously differentiable vector field in R 2, i.e. Φ = Φ 1 i + Φ 2 j. Then ( Φ2 x Φ ) 1 dx dy = Φ dr. (3.8) y Proof. ee Handout or [Bourne, pp ]. Remark 3.8. Green s Theorem in the plane is sometimes also referred to as tokes Theorem in the plane (e.g. in [Bourne, pp ]). orollary 3.9. The area bounded by a simple, closed, piecewise smooth curve R 2 is given by 1 2 ( yi + xj) dr. Proof. Apply Green s Theorem in the plane with Φ 1 (x, y) = y and Φ 2 (x, y) = x. 3.3 tokes Theorem Definition (a) A closed curve R 3, is simple, if it does not intersect itself. (b) A surface R 3 is orientable, if a unique normal can be assigned at each point x. Example. A Möbius strip for example is not orientable: Pfrag replacements P (c) Let be an open, orientable surface with simple boundary (curve). Let ˆn be the unit normal on. Imagine a person walking along the curve (in the positive direction) with its head pointing in the direction of ˆn. 34

6 ˆn Pfrag replacements L R Then and are said to be correspondingly orientated, if the surface is to the left of the person. [Anton, p. 1154], [Bourne, p. 210]. Theorem 3.11 (tokes Theorem). Let R 3 be an open, orientable, piecewise smooth surface with correspondingly orientated, simple, piecewise smooth boundary (curve) R 3. uppose that the vector field F is continuously differentiable (in a neighbourhood of ). Then ( F ) d = F dr. (3.9) Proof. ee Handout or [Bourne, pp ]. Remark (a) tokes Theorem implies that the flux of F through a surface depends only on the boundary of and is therefore independent of its shape. In other words, ( F Pfrag ) d replacements is the same for 2 1 and for (b) Note that Theorem 3.7 is a special case of Theorem To see this, assume that in Theorem 3.11 is flat, i.e. R 2 {0}. Then 35