MTH 255 VECTOR CALCULUS II SAMPLE PROBLEMS FOR MIDTERM I 0. Review your class notes and work through the HW problems! 1. Let c be the curve with vector equation r(t) = et sin ti + et cos tj. a) Find the arc length function s = s(t) for c measured from t =
Solutions to Math 20E Final Winter 99, Lindblad. 1. Adding the two equations together gives 3x y + 2 = 0 so y = 3x + 2. Inserting this in the rst equation gives z = 3y 2x + 1 = 3(3x + 2) 2x + 1 = 7x + 7. Hence the equation of the plane is (x, y, z ) = (t,
Solutions to MATH 20E 1-5 only.
FINAL EXAM Fall 98, Lindblad, Problems
1. (a) f = 6xyex y2 i + 3x2 ex y2 j so it increases most in the direction 12i + 3j. b) z z0 = fx (x0 , y0 )(x x0 ) + fy (x0 , y0 )(y y0 ) so z 3 = 12(x 1) + 3(y 2). 2. a) b)
FACT SHEET FOR 20E FINAL
1. From Chapter 8 The Divergence Theorem in 3D says that: where S = is the closed surface that bounds the domain R3 . Here n points to the exterior of . The Greens Theorem says that:
F n dS =
F dxdydz ,
(x F2 y F1 ) dxd
FACT SHEET FOR 20E EXAM 2
1. From Chapter 6 If (x, y ) = (u, v ) is a map between two domains and () in R2 , then the change of variables formula for integrals is: f (u, v )
dudv = (u, v )
f (x, y ) dxdy .
is the absolute value of the det
FACT SHEET FOR 20E EXAM 1
1. From Chapter 1 Dot product of two vectors v = (v1 , v2 , . . . , vn ) and w = (w1 , w2 , . . . , wn ): Cross produce of two vectors in 3D (its a vector!): v w = v1 w1 + v2 w2 + . . . + vn wn .
We have that v (v w) = w (v w) =
MATH 20E FINAL Please answer the following questions. You will not get credit for answers unless you demonstrate how you arrived at them. In short, please show all work. Numerical integrations by calculator will not be accepted.
Problem #1 ( pts.) Conside
EXAM # 2
Please a nsv/er t he f ollowing q uestions. Y ou w i,il n ot g et cred'it for answers unIess y ou d emonstrate h ow y ou a rrived a t t hem. I n s hort, please s how a ll w ork. Numerical i ntegrations b y c alculator w ill n ot b e a
MATH 1 10 EXAM # 2
Please answer the following questions. You will not get credit for answers unIess you demonstrate how you arrived at them. In short, please show all work' Numerical integrations by calculator will not be accepted.