Below, several regions R are shown. (a) Decide whether to use polar coordinates or Cartesian (rectangular) coordinates and then (b) Write | f(x, y) dA as an iterated integral, where f(x, y) is an arbitrary continuous function on R. (3.3) (-3,-3)
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- Find a parametric function for the intersection of r – 3y + 4z = 0 and a? + y² = 9.The vector function r(t) (1+2 cos t) i+ 3j+(5 – 2 sin t) k traces out a circle in 3-space as t varies. De- termine the radius and center of this circle.Let F = (7z+7x4) 7 + (3y + 2z + 2 sin (14)) 7 + (7x + 2y + 3e²¹ ) k. (a) Find curl F. curl F = (b) What does your answer to part (a) tell you about F. dr where C is the circle (x - 20)² + (y - 15)² = 1 in the xy-plane, oriented clockwise? JcF. dr = (c) If C is any closed curve, what can you say about F. dr? F. dr = (d) Now let C be the half circle (x - 20)² + (y − 15)² = 1 in the xy-plane with y > 15, traversed from (21, 15) to (19, 15). Find F. dr by using your result from (c) and considering C plus the line segment connecting the endpoints of C. JcF. dr =
- Find the integral fhpæp(z+6)/(z-6) by transforming to polar coordinates. I = %3DConsider the region R in the xy-plane bounded by (x2 + y2)2 = 9(x2 − y2). Convert the equation to polar coordinates. Use a graphing utility to graph the equation.Let ♬³ = (3z + 3x³¹) i + (4y + 3x + 3 sin (y¹)) j + (3x + 3y + 4e³¹) k. (a) Find curl F. curl F= (b) What does your answer to part (a) tell you about fc F. dr where C is the circle (x - 15)² + (y - 35)² = 1 in the xy-plane, oriented clockwise? Sc F.dr = (c) If C' is any closed curve, what can you say about foF.dr? Sc F.dr = (d) Now let C' be the half circle (x - 15)² + (y - 35)² = 1 in the xy-plane with y > 35, traversed from (16,35) to (14, 35). Find √F. dr by using your result from (c) and considering C plus the line segment connecting the endpoints of C. Sc F. dr =
- Q. Let f(z)= u(xy)+ Ż V(Z>Y) be analytic function and Z= re Find the Cauchy-Reimann equations in polar Coordinates ?Let F = (3z + 3x²) i + (6y + 4z + 4 sin(y²)) 3+ (3x+4y+6e²²) k (a) Find curl F. curl F = (b) What does your answer to part (a) tell you about SF. dr where C is the circle (x − 15)² + (y − 10)² = 1 in the xy-plane, oriented clockwise? ScF·dr = 0 (c) If C is any closed curve, what can you say about ScF. dr? ScF-dr = 0 (d) Now let C be the half circle (x − 15)² + (y – 10)² = 1 in the xy-plane with y > 10, traversed from (16, 10) to (14, 10). Find SF. dr by using your result from (c) and considering C plus the line segment connecting the endpoints of C. ScF·dr = 0Let w = f(x, y) and a, y in polar cordainate then (f.)² + (fy)² = %3D