Suppose F(2, y) = -y sin(æ)i + cos(2)j. (a) Find a vector parametric equation for the parabola y = x? from the origin to the point (4, 16) using t as a parameter. F(t) = (b) Find the line integral of F along the parabola y = x? from the origin to (4, 16).

Q: Evaluate the line integral / Vxy° +3 dx +6xy" dy, where C any curve joining (1, -1) to (16, 1).

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Q: The equation of the tangent plane to this ellipsoid at (1, –2, –2) is The vector parametric form of…

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Q: a. Find a parametrization of the curve x = (y* – 3)e2y.

A: We’ll answer the first question since the exact one wasn’t specified. Please submit a new question…

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A: Since we only answer up to 3 sub-parts, we’ll answer the first 3. Please resubmit the question and…

Q: (b)Find the derivative of g in the direction of a= 2i+ curl w where at point (0,-1,1) g = In- 4 + 2…

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A: According to Bartleby q&a guidelines I will attempt only one question

Q: Suppose F(x, y) = -y sin(x)i + cos(x)j. (a) Find a vector parametric equation for the parabola y =…

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

A: We can solve this as follows:

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Q: The velocity field for a fluid is given by V=(X2t3)i+(2XYT)j. The convective acceleration in the…

A: Given, V=x2t3i^+2xytj^ Here, u=x2t3 and v=2xyt Convective acceleration in y-direction is given by:…

Q: Suppose F(x, y) = -y sin(x)i + cos(x)j. (a) Find a vector parametric equation for the parabola y =…

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Q: Given: vector-valued function R(t) whose derivative R'(t) = (4t, 6t, —12t) (b) Reparametrize R using…

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Q: Q1 Use the parametric equations to calculate the line integral . ds over the path c which is given…

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Q: Use a scalar line integral to find the length of the curve r(t) = (20 sin , 20 cos ,5), for 0 < t <…

A: Given

Q: 3. Consider the curve C defined by the vector-valued function R(t) = -2):+-3)j+ cos(t – 2)k, - oo…

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Q: Sketch the direction field of the D.E. y = y - t and the solution curves corresponding to y(0) =-1,…

A: Solution of question is written in below steps.

Q: Suppose F(x, y) = -y sin(x)i + cos(x)j. x? from the origin to the (a) Find a vector parametric…

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Q: Show that the displacement u(x, t) in an elastic string of length t fireed at both ends which is set…

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Q: Suppose F(x, y) = -y sin(x)i + cos(x)j. (a) Find a vector parametric equation for the parabola y =…

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Q: An object rotates with angular acceleration, a, defined as dw a = dt where w is the angular…

A: Solving two ordinary differential equations, we get the result.

Q: Evaluate the line integral (e*y+ 2x) dx + (e* + 1) dy, where C is a smooth curve from the point (0,…

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Q: Given the scalar function f(x,y)=2xy,calculate the line integral along the ellipse curve from point…

A: Given data:- The given curve is, x29+y24=1 The given scaler function is, f(x,y)=2xy t E 0,π/2

Q: Use Green's theorem in order to compute the line integral (2sin z + 3y²) dx + (sin(4®) + 2x³) dy…

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Q: The parametric equations of a curve are given by x = In (VE +1) and Vt + y = cosh-(2t). X = dy when…

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Q: Suppose F(x, y) = (5x – y)ỉ + xj. %3D (a) Find a vector parametric equation for the line segment…

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Q: Suppose F(x, y) = -y sin(x)i + cos(x)j. Find a vector parametric equation for the parabola y = x²…

A: The solution is following.

Q: Consider the vector-valued function r(t) = (sin 2t + cos 2t)i + (sin 2t – cos 2t)j. (a) Find the…

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Q: 10. A particle moves in the xy-plane so that the position of the particle is given by x(1) =1+cos t…

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Q: | (xy dx + 2y dy)

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Q: Find an equation for the tangent plane to the graph of z = xe' cos z – 2 - at the point (2,0,0). 1.…

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Q: 1. Show that the problem may be formulated in terms of the velocity potential (r, 0) satisfying r dr…

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Q: 2. Find an equation for the plane tangent to the surface 3x2 - 4xy+ z2 = 0 at the point (1, 1, 1).…

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Q: Given the scalar functionf (x, y) = 2xy, calculate the line integral along the ellipse curve ²² +²²=…

A: We have to find the line integral along the given curve.

Q: Suppose F(x, y) = -y sin(x)i + cos(x)j. (a) Find a vector parametric equation for the parabola y =…

A:

Q: 16. Sketch the cycloid r(t) = (t- sint, 1 - cos t) together with its tangent vectors at t = and .…

A: Solved the above problem.

Q: The equation of the tangent plane to this ellipsoid at (1, –2, –2) is The vector parametric form of…

A: Let's find.

Q: 4. Consider the closed curve C, which is the image of the path 7(t) = (a cos t, b sin t), t E [0,…

A:

Q: Suppose F(x, y) = x²i + yj and C is the line segment segment from point P = (2, –1) to Q = (4, –4).…

A: We need to evaluate integral.

Q: Suppose F(x, y) = -yi + xj and C is the line segment from point P (3,0) to Q = (0, 2). (a) Find a…

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Q: Example 4. Find the tangent vector at the point (2,4). on the top half of the circle x² + y² = 1…

A: Introduction: For a given vector curve rt, the equation that represents the tangent line to the…

Q: Suppose F(x, y) = -y sin(x)i + cos(x)j. (a) Find a vector parametric equation for the parabola y =…

A:

Q: Assuming that the equations define y as a differentiablefunction of x, use Theorem 8 to find the…

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Q: 3 2 Use a line integral to calculate the length of the helix ř(t) = (a cost, a sint, bt) 0<t < 2n…

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Q: Use the Fundamental Theorem of Line Integrals to find the exact value of the path-independent line…

A: First of all hare we check for the path independent line integral then solve for moving over the…

Q: Suppose that over a certain region of space the electrical potential V is given by V (x, y, z) = 6x²…

A: We have to find the rate of change of V at point P in direction of v .

Q: (2x + y)dx + (x + 2y)dy

A: Given integral is:  where C is any curve that joins the point (1,1) with the (2,2).

Q: please answer the (a) (b) (c)  parts

A:

Q: Assuming that the equations define y as a differentiablefunction of x, use Theorem 8 to find the…

A: the equation we have  (x3-y4)6+ln(x2+y)=1

Q: Find the coordinates of the centroid of the curve x = cos t + t sin t, y = sin t - t cos t, 0…

A: Given,

Differential Equations

Have you ever observed the movement of the satellites and rockets or how the planets go around the sun in their orbits? How will you determine their motion? Definitely they follow some scientific laws and these kinds of problems are framed as differential equations.