MA 242 Test 3 Sample Last Name First Name Student ID Seat Code 1 1 8 points Write the precise statement of the following theorems t1 dt g dt Curve Parameterization RC f dr R t2 where g f dr t1 t2 is the domain for the parameter t Surface Parameterization RS f ds RD g dA where g f ru rv RHR D is the domain for the parameters u v r1 Fundamental theorem of curve line integral Assume f 0 Then RC f dr g r2 where g f r1 r2 C Stokes theorem curve integral surface integral RC f dr RS g ds where g f C S RHR Assume C is closed Then Stokes theorem surface integral curve integral Assume f 0 Then RS f ds RC g dr where g f C S RHR Gauss theorem RS f ds RD g dV where g f S D outward Assume S is closed Then 1 2 1 2 points Draw the roadmap 2 3 1 5 points RC f dr where f hy x 2y xzi C cid 8 x y z x2 y2 1 z 1 x 0 y 0 cid 9 Oriented counter clockwise looking toward z direction a Sketch the curve C or the surface S and show the orientation State whether it is open or closed b Compute f or f State whether it is zero or not c State the name of the theorem to be used d Apply the theorem show as much as shown in the class 3 4 1 5 points RC f dr where f cid 10 y2 3y2 2xy xy cid 11 C cid 8 x y z x2 z2 1 y 0 cid 9 Oriented clockwise looking toward to y direction a Sketch the curve C or the surface S and show the orientation State whether it is open or closed b Compute f or f State whether it is zero or not c State the name of the theorem to be used d Apply the theorem show as much as shown in the class 4 5 1 5 points RC f dr where f cid 10 z2 2xy x2 2yz y2 2xz cid 11 C cid 8 x y z y2 z2 1 x 0 y 0 z 0 cid 9 Oriented counter clockwise looking toward x direction a Sketch the curve C or the surface S and show the orientation State whether it is open or closed b Compute f or f State whether it is zero or not c State the name of the theorem to be used d Apply the theorem show as much as shown in the class 5 6 1 5 points RC f dr where f cid 10 z2 2xy x2 2yz y2 2xz cid 11 C cid 8 x y z x2 z2 1 y 0 cid 9 Oriented counter clockwise looking toward to y direction a Sketch the curve C or the surface S and show the orientation State whether it is open or closed b Compute f or f State whether it is zero or not c State the name of the theorem to be used d Apply the theorem show as much as shown in the class 6 7 1 5 points RS f ds where f hxy yz zxi S cid 8 x y z y z2 y 9 State whether it is open or closed a Sketch the curve C or the surface S and show the orientation 0 x 2 cid 9 The normal has negative y component b Compute f or f State whether it is zero or not c State the name of the theorem to be used d Apply the theorem show as much as shown in the class 7 8 1 5 points RS f ds where f hxy yz zxi S cid 8 x y z x2 y2 z2 1 cid 9 The normal is outward a Sketch the curve C or the surface S and show the orientation State whether it is open or closed b Compute f or f State whether it is zero or not c State the name of the theorem to be used d Apply the theorem show as much as shown in the class 8 9 1 5 points RS f ds where f hx y 0i S cid 8 x y z x y2 z2 x 9 cid 9 The normal has negative x component a Sketch the curve C or the surface S and show the orientation State whether it is open or closed b Compute f or f State whether it is zero or not c State the name of the theorem to be used d Apply the theorem show as much as shown in the class 9 10 1 5 points RS f ds where f hx y 0i S cid 8 x y z x2 y2 z2 1 cid 9 The normal is outward a Sketch the curve C or the surface S and show the orientation State whether it is open or closed b Compute f or f State whether it is zero or not c State the name of the theorem to be used d Apply the theorem show as much as shown in the class 10
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