Let f : R²\{0} = {x € R² : x # 0} → R be given in polar coordinates by Cos 20 f(r, 0) = r2 Sketch a few level curves in xy plane.
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Q: Consider the polar curves C1 : r = 3 sin 20 and C2 : r = 3 cos 0, and let R be the shaded region as…
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Q: 18. Show that the centroid of the sector in Figure 13 has y-coordinate 2R sin o (0, y) R FIGURE 13
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Q: 2. Let R(t) = sin 4t î – bcos"(21) j + 2t k. Find the moving %3D trihedral of the curve of R(t) at…
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Q: 3. Let f(x, y) = xy+ y². Find the equation of the tangent line in parametric form in the x-direction…
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A: The graph of the parametric curve x=√t, y=sint , 0≤t≤2π is given as below,
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Q: In changing the differentiating variable to a polar coordinate system, A. dx anddy becomes dr and…
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Q: 3.4.4 By transforming to polar coordinates, show that the double integral (x² + y²) ² dx dy (xy)²…
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Q: Suppose that f = u+ iv is analytic in a domain D. Show that the Cauchy-Riemann equations in polar…
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Q: Evaluate the line integral I of the one-form w = xy dx + 12y dy over the parametric curve a : [0,…
A: Given: ω=xydx+12ydy To find: I=∫αω where α:[0,ln2]→ℝ2 with α(t)=(e2t,e-2t) ∴x=e2t,y=e-2t Now,…
Q: Consider the polar curves C₂: r = 4 cos 0 and C3 : r = 4 sin 0. (a) Determine the slope of the…
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Q: Example 14. Use polar coordinates to evaluate the double integral 16-z2 dy dx (9 + x2 + y²)³/2 ° I
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Q: find the decomposition of a(t) into tangential and normal components at the point indicated r(0) =…
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Q: Question 9. Let N = {1< x² + y² < 4, x < 0} be a domain in R². Use Green's Theorem to evaluate…
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Q: 1) Use Green's theorem to evaluate o y'dx-x°dy where C is the curve shown: r2 where ri=3 and r2=5…
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Q: Consider the polar curves C2 :r = 4 cos 0 and C3 : r = 4 sin 0. %3D (a) Determine the slope of the…
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Q: Consider the polar curves C₂: r = 4 cos 0 and C3 : r = 4 sin 0. (a) Set up the integral that gives…
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Q: Q1: convert the Integral So √1-x² (x² + y²)dydx to the polar coordinates. 277
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Q: Suppose that f = u+ iv is analytic in a domain D. Show that the Cauchy-Riemann equations in polar…
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Q: Suppose that f = u + iv is analytic in a domain D. Show that the Cauchy-Riemann equations in polar…
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Q: (A) Find (jx curlū), grad (yzj • curlū) at น 7 = ({√Cos(1³² - x)) + ( x + ² } / + 20² k ze² where
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Q: Consider the polar curves C2 : r = 4 cos 0 and C3 : r = 4 sin 0. (a) Determine the slope of the…
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Q: Which one of the following is the parametric equa- tions of the line that is normal to the surface…
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Q: Consider the polar curves C1 :r = 3 sin 20 and C2 : r = 3 cos 0, and let R be the shaded region as…
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Q: 38. Let r(t) = (1³ -1- 2)i + 4r* j. %3D (b) Find parametric equations for the tangent line to the…
A: Topic = Vector
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Q: Consider the polar curves C2 :r = 4 cos 0 and C3 : r = 4 sin 0. (b) Set up the integral that gives…
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Q: Which one of the following is the parametric equa- tions of the line that is normal to the surface…
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- Use Green’s theorem to evaluate ∮C(ye2xy−5y)dx+ (xe2xy−2x)dy, where Cis the counterclockwise oriented boundary curve of the square with vertices at(0,0), (0,1), (1,0), and (1.1).Let F = (-z2, 2zx, 4y - x2}, and let C be a simple closed curve in the plane x + y + z = 4 that encloses a region of area 16 (Figure 20). Calculate ∮C F • dr, where C is oriented in the counterclockwise direction (when viewed from above the plane).Use Green's Theorem to evaluate∮tan^-1(y)dx-(xy^)/(1+y^2) dy where C is the square with vertices (0, 0), (1, 0), (1, 1) and (0, 1) and oriented counterclockwise. A. -1 B. 2 C. 1 D. -2
- Use Greens Theorem to evaluate the line integral ∫C(x2-2xy)dx + xy2dy where C is the positively oriented curve determined by y2 = x and y = −x from (0, 0) to (1, −1). Include a sketch of the curve.Let u(x,y) satisfy the following equationuxx+(x2+y2-1)uyy=0Find the region in which the equation is elliptic and the region in which the equation is hyperbolic.Evaluate the flux (F*n)d-sigma where F = (3xy^2, 3x^2y, z^3) and M is the surface of the sphere of radius 10 centered at the origin.
- Find the area enclosed by one loop of this polar curve: r=3sqrt(cos2theta) from 0 to 2pi using the formula A=1/2 integral from 0 to 2pi (r)^2 for parametric curve.Let C be the curve of intersection of the spheres x 2 + y 2 + z2 = 3 and(x - 2)2 + (y - 2)2 + z2 = 3. Use the result of Exercise 63 to find parametric equations of the tangent line to Cat P = (l, 1, 1).Consider the surface x2 + y2 − 2xy − x + 3y − z = −4 and the point P(2, −3, 18). Determine the the parametric equations of the normal line to the surface at P0