Don't worry, you'll see what I mean in the next article. 21.3 FLUX INTEGRALS OVER PARAMETERIZED SURFACES Most of the flux integrals we. In each of the following cases, how does the flux change if the. A change of coordinates (, ), (, ) where the rectangle 0 1. ![]() (3) If C is a simple closed curve, what is. (2) Use Greens Theorem to evaluate the line integral (In x + y) dx x2 dy over the rectangle in the xy-plane with vertices at (1, 1), (3, 1), (1, 4), and (3, 4). An integral transport theory method is described for calculating the flux distribution in an infinite, rectangular lattice, assuming mono-energetic neutrons. It could be in front of the xy-plane, or the yz-plane or the xz-plane. The integral SF dA represents the flux of F through this rectangle, S, oriented upward. (1) Use Greens Theorem to evaluate the line integral xy dx + y dy where C is the unit circle orientated counterclockwise. ![]() ∮ C F ⋅ n ^ d s = ∫ 0 2 π F ( r ( t ) ) ⋅ ⏟ Velocity at a point n ^ ( r ( t ) ) ⏞ Normal vector at a point ∣ ∣ r ′ ( t ) ∣ ∣ d t ⏟ d s = ∫ 0 2 π ( ⋅ ) ∣ ∣ ∣ ∣ d t = ∫ 0 2 π ( ⋅ ) ∣ ∣ ∣ ∣ d t = ∫ 0 2 π ( 9 cos 3 ( t ) + 3 sin 2 ( t ) ) 3 2 sin 2 ( t ) + 3 2 cos 2 ( t ) d t = ∫ 0 2 π ( 9 cos 3 ( t ) + 3 sin 2 ( t ) ) 3 sin 2 ( t ) + cos 2 ( t ) ⏟ = 1 d t = ∫ 0 2 π ( 9 cos 3 ( t ) + 3 sin 2 ( t ) ) 3 d t = 9 ∫ 0 2 π ( 3 cos 3 ( t ) + sin 2 ( t ) ) d t \begin C start color #bc2612, C, end color #bc2612 itself, since that's all you need. The region D coud be rectangular, or any region. The function (u,v)(ucosv,usinv,v) parametrizes a helicoid when (u,v)D, where D is the rectangle 0,1×0,2 shown in.
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