Compute for the given oriented surface.
F = <ez, z, x>, G(r, s) = (rs, r + s, r), 0 ≤ r ≤ $a, 0 ≤ s ≤ $b, oriented by Tr × Ts
Recall how to find vector surface integrals. Let F be a vector field. Let G(u, v) be an oriented parametrization of an oriented surface S with parameter domain D. Assume that G is one-to-one and regular, except possibly at points on the boundary of D. Choose the correct integration for
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
If the orientation of S is reversed, the surface integral changes sign.
Find Tr and Ts for G(r, s) = (rs, r + s, r). Then use these vectors to compute n(r, s).
Tr = (∂G / ∂r)
= <MsZ/bYLXueA=, 0VV1JcqyBrI=, 0VV1JcqyBrI=>
Ts = (∂G / ∂s)
= <xn4vb6syll8=, 0VV1JcqyBrI=, 1Wh3cvJ2xF4=>
n(r, s) = Tr × Ts
= <UYinCekNO0E=, xn4vb6syll8=, 4+12+vmTTkofKOSw>
Compute F(G(r, s)) given that F = <ez, z, x> and G(r, s) = (rs, r + s, r).
F(G(r, s)) =
bg/567bfyVupZS+eRuuviSmy1WmIysKJ97embUCp+vJDs2fBgANUUlj4w1toLwd/iEATfiJ47cj1SXZbRxXCvXko2PM3eI/YNSI11F/PAU3T8CF3/27yEKrB5a7MdeeyfA/zIRREgl3D3s9Q8SaFg9ZJM8GYZAyzg9wSVGq+tUMk716EoBZHocDz55HVG8R4bqD1W2du1v9g6QUgfmGSwAIXZQTbK8tQr1LseXSOyRfObl30ce6Ay3ryK5HeDFimaJOhugTQxem9qXn7R+lBvGGhRPJmfXOc990dO8lkzU66J0hoSWJEoxxZ6YU9JpMIndJQGUeIMngFuKaEoEvFTdq8EDuRI9l8ACteyabNuH9+J84F1f77I9DR4xw4QM57ymp3g1OHCyuDPG8QHJ5L+PeIERvJwig/RywkEc2D+XtR3H4OY/0t8OlQItwVPjIchSp6SdOVDnYvk5wjhmoOD0XSAmMxo4G2bLRW73gTPEGqWAu9cJ19GvwfvA6Kx9Xhht+G7S5mVD4ViQugpIOpt5mhmMGNsEoA02SSZnOjeYs22adSuRBMxQ/A0Py/cJiAKKri1fyRMJ9Ju2H2Klbbkw6Z/5C6nfQ0/IzDZDofXZDK0rsRFJLdBkyHpv0JeSbt4zCo9XbZ2bjiBgGB8AYcwhSiEbinXS5oI+rSwEB8OkgSmTyhd0qFBConlzNsS8LcijISloaQQMwa2Ok/WacKjyH4l1HVoXGd8o0G1vmL72FkH+I0Spb6YkZCAz9fSJG+B1bnsCEqNI3HnAeLFHbIBRFWchbeJ8eTLtVb4onMxidFewdvpRSqM5rWTFGUY65v6ZKGzYuCOj6mituAB8L1YcLlF2iFxT0jteUZ6JELVegPwKZvswqa3azsMO33iEFYUefi9ES7mN9Cq62jEhYAMFW9jEE0v2Zba0BVlOy1EmlcMBm/pjOCkg2f3gs2d9Bxq/VUU/CAVfzw3slKNuA2YBnWeVr98IzdqR+mkYESJZhodqcNa71xPR1OLMrewgY+dnMY2I8oKS2UIvT60n+VSOomg/47h5fBjeTy1Rb6+2wxKENQ+uTVKA==Compute the dot product F(G(r, s)) · n(r, s).
F(G(r, s)) · n(r, s) =<er, r, rs> · <-1, r, s - r > =
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Compute using the restrictions 0 ≤ r ≤ $a and 0 ≤ s ≤ $b. Since we are orienting by Tr × Ts, which is the normal vector we found in Step 1, we RWxeBpWG+dilGrB2blwCC/qjz2U= need to change the sign of the integral.
= iSba6t70dtA=(zDsgB75cNfg= - e$a) - bEz/iYnXpnk=