Let F= 3xyi + 2x² j and suppose C is the oriente curve shown in Figure 28. Evaluate F.ds both directly and also by means of Green's theorem. y (0,0) (2,0) 2) (2,-2 )

Q: Let y be the curve with vector form r(t) = (1/(t+ 2), sin(sin(sin t)), e-"). a) Determine whether…

A: The parametric curve is r(t)=⟨1t+2, sin( sin( sin t)),e-t⟩. x(t)=1t+2, y(t)=sin( sin( sin t)),…

Q: Let 2ry + T sin y = 27. a) Verify that the Po(1, 7/2) is on the given curve. b) Write down equations…

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Q: Show that cos(wt – B), cos wt , sin wt are linearly dependent functions of t.

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Q: 1. If the parametric curve $x=f(t), y=g(t)$ satisfies $g^{\prime}(1)=0,$ then it has a horizontal…

A: TRUE Slope of the curve, when in parametric form = dy/dx = g'(t) / f'(t)

Q: Let C be a smooth curve traced by Ř(t) with R(1) = (2, –1, 1) and R' (1) = (1, –2, –1). Let g(t)…

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Q: et C be a portion of a curve from (2,0) to (2,2). Let F(r, y) = (tan(2?) +2, ) Is F conservative?…

A: In the given question, the concept of integration is applied. Integration Integration is a method of…

Q: Show that if the function f(x, y) = ex siny + ecosx, satisfies Laplace Equation.

A: We can solve this as follows:

Q: Let C be a portion of a curve from (2,0) to (2,2). Let F(r,y) = (tan(x²) + e²v¸ 2re²v

A: Introduction: A vector field is conservative if ∂P∂y=∂Q∂x, Where f(x,y)=Pi+Qj.

Q: Calculate F.dr, where C is the curve given by ti + 123+ tk, with t = [0, 1] and F(x, y, z) = 4xei +…

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Q: Sketch the graph of 7(t) = sin(t)i – 2cos(t)j–tk in 3-space; then find and draw the tangent vector…

A: Given, r→t=sinti^-2costj^-tk^

Q: Find an equation for the tangent plane to z = cos(x2 + y2) at the point /等,0).

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Q: Show that 22 dz = 0 1 – cos(2) whenever y is a closed simple smooth curve contained in a very small…

A: Solution: Here the integral is ∮+γz21-cosz We know that    cos0=1cos2π=1cos3π=1... Only z=0 is in…

Q: Sketch by hand the direction field for the following ODE: y' = = xy. In particular, use isoclines…

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Q: Let C be the curve defined by T(t) = (2 + 2 sin(t), 3 – 2 cos(t)) for 0 < t < 4x=²) dx + (2xyev" +…

A: Given the integral is ∫Cy2exy2+4x-2dx+2xyexy2+2dy. Rewrite the integral as ∫CF·→dr→:…

Q: Let C be the curve defined by 7(t) = (2+2 sin(t), 3 – 2 cos(t)) for 0 <t< L (e + 4z) dz + (2zye" +…

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Q: Let flxiy) = Rx Jy 27 %3D a) Finid the gradrent of f b) Find the directional derivative of f at the…

A: Consider the function as  fx,y=2xy

Q: Let f(x, y) = 2xy – 4x² + 3y². %3D Then a standard equation for the tangent plane to the graph of f…

A: The function is given. Find the partial derivative fx and fy at the given point. Given,…

Q: Find the tangent plane to the equation 3y cos(4x – 2y) at the point (2,4,12) Z =

A: Z=3ycos(4x-2y)  Point= (2,4,12)

Q: Let C be the curve given by X- Sin/tr) and y=t, from ĭst 55. sin/tT and y=t, from ist E5. Find the…

A: According to Fundamental Theorem of Line Integral, the definite integral of a function is evaluated…

Q: Use Green's Theorem to evaluate F. · dr, where F = (Va + 4y, 2x + and C consists of the arc of the…

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Q: Find Sc F · dr for F = 3yi – (sin y)j on the curve counterclockwise around the unit circle C…

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Q: Let C be the curve connecting (0,0,0) to (1,3,1) to (3,4,4) to (2,1,3) to (0,0,0) Evaluate + y)dx +…

A: Given,     and

Q: f(x, y) =(e-2y)Cos2x

A: The Laplace equation is: ∂2f∂x2+∂2f∂y2=0

Q: Find the vector equation of the tangent line to the curve given by 7(t) = sin ti + cos (2t)j + tan…

A: There is no such introduction for the solution.I have explained it within the solution.Please go…

Q: find a parametrization c(t) of the curve satisfying the given condition.  y = 3x − 4, c(3) = (2, 2)

A: To find the parametrization

Q: Let C be the curve parameterized as follows F(t) := (9 cos(t), 9 sin(t), 12t), 0<ts 10 and let f(x,…

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Q: 6. (a) Show that F = (3a²y+2, x3 + 3) is conservative. (b) Evaluate Sc(3x²y +2)dx + (x³ + 3)dy where…

A: According to the problem, we have

Q: Compute the tangent vector to the given path. c(t) = (2e", 2 cos(t))

A: The given path is: c(t)=(2et,2cos(t)) To find the tangent vector to the given path c(t).

Q: Let C be the curve consisting of line segments from (0, 0) to (1, 1) to (0, 1) and back to (0, 0).…

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Q: Consider the curve C in R3 whose parameterization is given by: eq. in image If is there a point on…

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Q: Compute the tangent vector to the given path. c(t) = (4t sin(t), 6t) %3D

A: The tangent vector for a vector r→t is r→'t.

Q: Evaluate F ds (or F ds) where Jc F = (y – x2, z - y², x – z) and C is the curve r = joining (0,0,0)…

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Q: (3)If F = (x points (2,1,1) ² yz, xyz² , zy² ). Find curl curl F at

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Q: Let C be the curve defined by r(t) [6 cos(t) – 5 cos (t), 6 sin(t) – 5 cos(t) sin(t), cos(t)] for 0…

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Q: Compute / f ds over the curve specified. (1) f(x, y, z) = z², r(t) = for 0 < t < 2

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Q: 13. Let F (x, y) = (1(²y²)x, af(x²y? , a(y²) Y) and suppose that C is a smooth curve parametrized…

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Q: Find a tangent vector of the curve7(t) = (t², 2 sin(t), 2 cos(t)) at (0,0, 2) (0,0,1) (1,0,0) None…

A: Here, Find it’s derivative This is the tangent at any point on the curve.

Q: Find the tangent plane for the level surface f(x,y,z) = 7x5 + cos (5y) + In (9z) = c at the point…

A: Answer: In oerder to use gradients we introduce a new variable c= 7x5+cos(5y)+ln(9z) Therefore the…

Q: 5. Verify Green's Theorem in the plane for F = (xy + y²)î + (x – y)f and the curve C is as shown in…

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Q: Show that if the function f(x, y) = ecos-e-cosx, satisfies Laplace Equation.

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Q: A curve is given by the parametrization r(t)= [²,-t, ť ]. If the tangent vector to this curve at a…

A: The given curve is,  ,

Q: Suppose that C is the graph of a smooth vector-valued function r(s) parametrized by arc length with…

A: Introduction: The tangent vector to a vector valued function rt is formulated by the equation r't…

Q: Find the arclength of the curve y=2-x3/2 from (0, 2) to (9,3).

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Q: Graph the curve. x = cos(t) + In(tan(t/2)), y = sin(t), A/4 sts 3/4 y y 1.2- -0.3 -0.2 -0.1 0.1 0.2…

A: Using graphing calculator we graph the function Answer(a):

Q: Let f(z) = z³. (a) Determine the curve(s) that can produce the image u = 0 under the mapping of f…

A: The objective is to solve the above question.

Q: Let curve C be the arc of x2 + y2 = 16 traversed once counterclockwise from (0, 4) to (0, -4). Let…

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Q: Consider the parametric curve segment (t, t2), t ∈ [0, 1]. What is the firstorder derivative of the…

A: We have the given parametric curve t,t2, t∈0,1x=ty=t2 for t∈0,1the first order derivative the curve…

Q: Let H = 5rsinø r - rz coso + 2rî . At point P(2,30°,-1), determine (a) the component of H normal to…

A: Given At point (2,30°,-1) To find  Component of H normal to r=2 and  Component of H tangential to…

Q: Consider a curve parameterized by: c(t)=(sin^3(t),cos(t)) Compute dy/dx at the point ((sqrt(2)/4),…

A: Differentiate the given function as follows.

Q: Suppose you are told that there does not exist a smooth space curve K de- scribed by a vector…

A: In this question, we need to check the validity of the arc length function. The given arc-length…