The polar equation r = θ defines a spiral in the xy-plane. Let C be the portion of this spiral starting at (x, y) = (−3π, 0) and ending at (x, y) = (0, 0). Let
F(x, y) = < (y − 1)^e cos(x−xy) sin(x − xy), xe^cos(x−xy) sin(x − xy) >
Find Z C F · dr
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The polar equation r = θ defines a spiral in the xy-plane. Let C be the...
(2) Let x-r cos θ, y-r sin θ represent the polar coordinates function f(r, θ) : R. R2, Compute f, (r$) and f, ( ompute * T
(2) Let x-r cos θ, y-r sin θ represent the polar coordinates function f(r, θ) : R. R2, Compute f, (r$) and f, ( ompute * T
The path integral of a function f(x, y) along a path e in the xy-plane with respect to a parameter r is given by 2. fex,y)ds= f(x),ye) /x(mF +y(t" dr , where a sr sb. (a) Show that the path integral of f(x, y) along a path c(0) in polar coordinates where r=r(0), α<θ<β, is Sf(r cos 0,rsin e) oN+( de. (b) Use this formula to compute the arc length of the path r 1+cos0, 0<0 27
The path integral...
A curve in polar coordinates is given by: r = 9 + 2 cos θ Point P is at θ = 20π/18 (1) Find polar coordinate r for P, with r > 0 and π < θ < 3π/2. (2) Find cartesian coordinates for point P (3) How may times does the curve pass through the origin when 0 < θ < 2π?
10. Stokes' Theorem and Surface Integrals of Vector Fields a. Stokes' Theorem: F dr- b. Let S be the surface of the paraboloid z 4-x2-y2 and C is the trace of S in the xy-plane. Draw a sketch of curve C in the xy-plane. Let F(x,y,z) = <2z, x, y?». Compute the curl (F) c. d. Find a parametrization of the surface S: G(u,v)- Compute N(u,v) e. Use Stokes' Theorem to computec F dr
10. Stokes' Theorem and Surface Integrals...
Name: Convert the polar equation to rectangular form. 11. r-iosin θ 78 12. Find the eccentricity of the polar equation r-26+cos θ.
Name: Convert the polar equation to rectangular form. 11. r-iosin θ 78 12. Find the eccentricity of the polar equation r-26+cos θ.
10. Stokes Theorem and Surface Integrals of Vector Fields a Stokes Theorem:J F dr- b. Let S be the surface of the paraboloid z 4-x2-y2 and C is the trace of S in the xy-plane. Draw a sketch of curve C in the xy-plane. Let F(x,y,z) = <2z, x, y, Compute the curl (F) c. d. Find a parametrization of the surface S: G(u,v)ーーーーーーーーーーーーー Compute N(u,v) e. Use Stokes' Theorem to compute Jc F dr
10. Stokes Theorem and Surface...
10. Stokes' Theorem and Surfac e Integrals of Vector Fields a. Stokes' Theorem: F-dr= b. Let S be th ky-plane. Draw a sketch of curve C in the xy-plane. et be the surface of the paraboloid z 4-x-y and Cis the trace of S in the c Let Fox.y.z) <2z, x, y>, Compute the curl (F) d. Find a parametrization of the surface S: G(u,v)- Compute N(u,v) F-dr Use Stokes' Theorem to compute , e.
10. Stokes' Theorem and Surfac...
(15 points) Find the centre of the region in the xy-plane that lies inside the cardioid r = a(1 + cos θ) and outside the circle r-a if the mass density is p(,y)-1
(15 points) Find the centre of the region in the xy-plane that lies inside the cardioid r = a(1 + cos θ) and outside the circle r-a if the mass density is p(,y)-1
Prob. 4 Assume that on the xy-plane, vectors P and Q make angles θ and φ with respect to the r-axis. Use the basic properties of the dot-product of vectors, show that cos(θ + φ)-cos θ cos φ-sin θ sin φ. Also, use the basic properties of the cross-product, show that sin (e+ φ)-cos θ sin o + cos θ sin o.
for the curve r(t) find an equation for the indicated
plane at the given value of t
56) r(t) (t2-6)i+ (2t-3)j+9k; osculating plane at t=6 A) x+ y+(z+9)=0 C)x+y+ (z-9)-0 56) B) z-9 D) z -9 (3t sint+3 cos t)i + (3t cos t-3 sin t)j+ 4k; normal plane at t 1.5r.. A) y=-3 57) r(t) 57) B) y 3 C)x-y+z-3 D) x+y+z=-3
56) r(t) (t2-6)i+ (2t-3)j+9k; osculating plane at t=6 A) x+ y+(z+9)=0 C)x+y+ (z-9)-0 56) B) z-9 D)...