Methods 2 Notes 2

Question 1

(2D-N41-) Closed curve

Answer 1

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Question 2

(2E-N41-2.2) The length of a curve C , parametrized by r : (a, b) 7to Rd is given by

Answer 2

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Question 3

(2E-N42-2.3) Find the length of the curve C defined by r : [0,4pi] to R3, with r(t) := 2/3 t^1.5 i + cos(t)j + sin(t)k

Answer 3

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Question 4

(2E-N43-2.4) Let C be the right half of the circle x2 + y2 = 16. Evaluate integral xy^4 dr over c

Answer 4

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Question 5

(2D-N44-2.5) Line integral (or path integral) of a vector field F over a curve C. 5 different ways to write this

Answer 5

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Question 6

(2E-N45-2.6) Let F (x, y) := yi + xyj and the curve C be given by r(t) := cos(t)i + sin(t)j for t between or equal to 0 and pi. We want to compute the line integral of F along C. Do this two ways

Answer 6

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Question 7

(2T-N46-) Properties of line integrals

Answer 7

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Question 8

(2R-N47-) Explain path independence

Answer 8

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Question 9

“(2E-N47-2.9) Con- sider the vector field F(x,y,z) = yi + xj. Fix any two points (a,b,c) and (d,e,f) in space and connect these two points with any piecewise smooth curve C. Let r(t) =
x(t)I + y(t)j + z(t)k be a parameterization of this curve so that r(0) = (a, b, c) and
r(p) = (d,e,f) for some positive real number p.”

Answer 9

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Question 10

(2R-N48-) If line integral over c is path independent, line integral over any closed curve is 0!

Answer 10

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Question 11

(2L-N48-2.10) If line integral over a closed path c is not 0, integral over c is not path independent. Plus useful note

Answer 11

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Question 12

(2R-N48-) Forumla for evaulataing double integral. Give type of function and nature of region as well

Answer 12

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Question 13

(2E-N49-2.10) Region -1 <= x <= 1, x^2 -1 <= y <= x^2 + 1. Let f from R^2 to R be defined by f(x,y) = 1

Answer 13

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Question 14

(2E-N49-2.11) Region 0 <= y <=1, 0 <= x <= -y +1. Let f from R^2 to R be defined by f(x,y) = xy^2

Answer 14

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Question 15

(2D-N51-) Jacobian, what is it for 2d, 3d, when use it

Answer 15

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Question 16

(2E-N52-2.12) Let’s begin with a simple example in which we already know the answers. Let R be the square with side length 1 centered at the origin and consider the function f (x, y) = 1. Evaluate DI, with transform u and v such that x(u, v) = 3u and y(u, v) = v+1

Answer 16

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Question 17

(2R-N53-) Polar coordinate transformation with Jacobian. What is it.

Answer 17

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Question 18

(2E-N53-2.13) Let f(x,y) = 3x + 3y and let the region of R be defined by x^2 + y^2 =<9, y>= 0 . Compute integral

Answer 18

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Question 19

(2E-N54-2.14) Let the region V be the ball of radius 1 centered at the origin, i.e., the set of (x,y,z) such that x^2 + y^2 + z^2 <= 1. (know which transformation of coordinates and how) (also know every type in 3d)

Answer 19

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