Study guide 3

Study guide 3 - MA 261 - Fall 2010 Study Guide # 3 You also...

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Unformatted text preview: MA 261 - Fall 2010 Study Guide # 3 You also need Study Guides # 1 and # 2 for the Final Exam i+ j+ k; if F(x, y, z) = P (x, y, z) i + Q(x, y, z) j + R(x, y, z)k, then x y z i curl F = F = x P Properties of curl and divergence: (i) If curl F = 0, then F is a conservative vector field (i.e., F(x) = f (x)). (ii) If curl F = 0, then F is irrotational; if div F = 0, then F is incompressible. (iii) Laplace's Equation: 2 f = 2f 2f 2f + + = 0. x2 y 2 z 2 j y Q k z R and div F = F = Q R P + + x y z 1. Del Operator: 2. Parametric surface S: r(u, v) = x(u, v), y(u, v), z(u, v) , where (u, v) D: n z v r(u,v) D u y S x Normal vector to surface S : n = ru rv ; tangent planes and normal lines to parametric surfaces. 3. Surface area of a surface S: (i) A(S) = D |ru rv | dA 1 + (h/x)2 + (h/y)2 dA; D (ii) If S is the graph of z = h(x, y) above D, then A(S) = Remark: dS = |ru rv | dA = differential of surface area; while dS = (ru rv ) dA 1 4. The surface integral of f (x, y, z) over the surface S: (i) S f (x, y, z) dS = D f (r(u, v)) |ru rv | dA. (ii) If S is the graph of z = h(x, y) above D, then f (x, y, z) dS = S D f (x, y, h(x, y)) 1 + (h/x)2 + (h/y)2 dA. 5. The surface integral of F over the surface S (recall, dS = (ru rv ) dA): F dS = S D F (ru rv ) dA. F dS = S S (F n) dS = D F (ru rv ) dA. If S is the graph of z = h(x, y) above D, with n oriented upward, and F = P, Q, R , then F dS = S D -P h h -Q +R x y dA. (i) Connection between surface integral of a vector field and a function: F dS = S S (F n) dS. (The above gives another way to compute (ii) S S F dS) F dS = S (F n) dS = flux of F across the surface S. n S 6. Stokes' Theorem: C F dr = S curl F dS (recall, curl F = F). n S C F dr = circulation of F around C. C 2 7. The Divergence Theorem/Gauss' Theorem: S F dS = E div F dV (recall, div F = F). n S n E n n 8. Summary of Line Integrals and Surface Integrals: Line Integrals C : r(t), where a t b Surface Integrals S : r(u, v), where (u, v) D ds = |r (t)| dt = differential of arc length dS = |ru rv | dA = differential of surface area ds = length of C C S dS = surface area of S b f (x, y, z) ds = C a f (r(t)) |r (t)| dt S f (x, y, z) dS = D f (r(u, v)) |ru rv | dA (independent of orientation of C ) dr = r (t) dt (independent of normal vector n) dS = (ru rv ) dA b F dr = C a F(r(t)) r (t) dt S F dS = D F(r(u, v)) (ru rv ) dA (depends on orientation of C) F dr = C C (depends on normal vector n) F dS = S S F T ds F n dS The circulation of F around C The flux of F across S in direction n 3 9. Integration Theorems: b Fundamental Theorem of Calculus: a F (x) dx = F (b) - F (a) a b b Fundamental Theorem of Calculus For Line Integrals: a f dr = f (r(b))-f (r(a)) C r(a) r(b) Green's Theorem: D Q P - x y dA = C P (x, y) dx + Q(x, y) dy D C Stokes' Theorem: S curl F dS = C F dr n S C Divergence Theorem: E div F dV = S F dS n S n E n n 4 ...
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