Suppose F(x, y) = -y sin(x)i + cos(x)j. Find a vector parametric equation for the parabola y = x² from the origin to the point (4, 16) using t, varying from 0 to 4, as the parameter. F(1) = (t, f(t)) = Find the line integral of F along the parabola y = x² from the origin to (4, 16).

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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: Exercise 10.6 Evaluate the line integral using a method of your choice. e¯*(cos y dx + sin y dy),…

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

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Q: Find the linearization L(x,y) of the function f(x,y) = e 3× cos (9y) at the points (0,0) and 0,

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Q: 1. Give an equation of the tangent plane to the surface z = In(4y – x) at the point (3,1,0). 2. Use…

A: Since you have asked multiple questions, we will solve the first question for you. If you want any…

Q: Find the equation of the tangent plane to the surface z = ln (2x + y) at the point (-1,3,0) .…

A: To find the equation of the tangent plane to the surface z=ln2x+y at the point -1, 3, 0.

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: 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: Suppose F(x, y) = -yi + xj and C is the line segment from point P = (5,0) to Q = (0, 3). (a) Find a…

A: Given: F→(x, y)=-yi→+xj→For part(d)we know∫TF→.dr→=∫C1F→.dr→+∫C2F→.dr→+∫C3F→.dr→where T is the…

Q: Evaluate the line integral f. F. dr where F = (2 sin a,–5 cos y, 10xz) and C is the path given by…

A: We will evaluate the given line integral.

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: 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: (a) Suppose that a black hole has magnetic field F = (2a* + e" sin(2), y³ + xe*, 2z³ + e" cos(z) +…

A: TO FIND: Magnetic flux across the black hole

Q: Find the equation for the Tangent Plane on the given surface coS ax – x²y + e*z+ yz = 4 Po(0, 1, 2)

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Q: If F(x, y) = (6xy - 1,3x2 + 2y) and C is the circular arc along x² + y? = 2 from (1, 1) to (-1, 1),…

A: Given vector field is  F(x,y)=6xy-1,3x2+2yF(x,y)=(6xy-1)i+(3x2+2y)j c is the circular arc along…

Q: 4. Let f(x, y) = sin(x²y) + vx + e+y. Find the equation of the tangent plane to the surface z f(x,…

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Q: Evaluate the unoriented line integral (2æ² + y² + z²) ds along the parametric curve a : [0, 1] → R³…

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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: 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: 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: 4. Integrate f(x, y, z) = Vx² + y² over the involute curve (cos t + t sin t)i + (sin t – t cos t)j,…

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

A: Given:           x3−2y2+xy=0,     (1,1)

Q: Show that the function f(x, y) = In /x² + y² satisfies the Laplace equation. %3D

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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: C3x-3y2 +2x2) dxt (3y-3x2+2²)dy (3²-3xytx*2yz)da calculate the curvilinear integral.where cis any…

A: Let's find.

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: Let C denote the curve of intersection between x = -5z° and y = 6z", starting from A(0,0,0) and…

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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: 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: Suppose F(2, y) = -y sin(æ)i + cos(2)j. (a) Find a vector parametric equation for the parabola y =…

A: Given F→(x,y) = -ysin(x)i→+cos(x)j→ a) Let us assume x=t then y=x2=t2 by differentiating r→(t) with…

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: Consider the surface x°z5 + sin(y4z5) – 8 = 0. - Use implicit differentiation to find the following…

A: SOLUTION-

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

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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

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Q: if f(x,y) =x2y÷x+y and y(t)=  (x(t),y(t)) is parametrized curve in domain of f with y(0)=(2,-1) and…

A: The first step in solving the problem is to obtain the parametric equation. The initial values can…

Q: 1. Evaluate the line integral I = = [v¢4x° + y*) dx + x(2x° + 3y*) dy] around the ellipse x/a? +…

A: The given line integral is, I=∮y4x2+y2dx+x2x2+3y2dy We know the Green's theorem:…

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