8. Change to cylindrical coordinates and find the divergence and curl of (a) F xi + yj x² + y² -yi + xj x² + y² (b) F =

Q: (a) Find a vector parametric equation for the parabola y = x² from the origin to the point (2, 4)…

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Q: Which of the space curves below describes the vector-valued function r(t) = (t + 15, e0.08t cos t,…

A: The given parametric equation of the curve is  where  is the parameter.To determine which of the…

Q: Let C be the curve of intersection of the paraboloid x² + y² = z and the cylinder + (y − 1)² = 1 (a)…

A: We will use the basic knowledge of vector calculus and differential calculus to answer all three…

Q: Given w f(z,y, 2) = zy²z. Find: Ograd/(1,3,-2) 3. (b) equation of the tangent plane to the surface…

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Q: Find the set of parametric equations for the tangent line to the graph of the vector-valued function…

A: To find the set of parametric equations for the tangent line to the R(t)=<t+sin3t, t+cos3t,…

Q: 4. Let r(f) = ° 13 i - 6fj + Pk. (a) Find the unit tangent vector T(t) and T(3). | (b) Find the…

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Q: Find the equation of the tangent line to the trace y-3 of the quardic surface defined by 9x² + 2y²…

A: We have to find the equation of the tangent line to the trace y=3 of the quardic surface defined by…

Q: a) By first sketching its projection onto the yz-plane, sketch the curve r(t) = (t, cos(t), 2…

A: For (a), We will plot the sketch of r(t)=t, cos(t), 2 sin(t),  0≤t≤4π onto the yz-plane (that means…

Q: Ř(u, v) = (u², v – 2u, u + v²)

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Q: 1. Let R and b be positive constants. The vector function r(t) = (R cost, R sint, bt) traces out a…

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Q: 4. Consider the vector function r(z, y) = (x, y, x² + 2y*). (a) Re-write this vector function as…

A: For the given vector function, PART A The surface function is given by, f(x,y)=xi^+yj^+(x2+2y2)k^…

Q: Sketch the parametric curve of r(t) =.Then, find and draw the tangent vector r'(0)

A: 2cos t, 2sin t,3To Find:a. Sketch the parametric curve.b. Find and draw the tangent vector r'(0).

Q: Find an equation of the tangent plane to the parametric surface

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Q: (a) Find a vector parametric equation for the ellipse that lies on the plane x + 3y + z = 5 and…

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Q: Find a vector parametrization of the circle of radius 5 in the xy-plane, centered at the origin,…

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Q: 1. Obtain the directional derivative of: a.. f(x,y) = x²-4x³y² at the point (1,-2) in the direction…

A: Step 1:

Q: t-) For each surface below, find the normal vector and equation of the tangent plane at the given…

A: Use definition of tangent plane and norman vector.  According to guidelins, we solve only first 3…

Q: 7. For the vector-valued function √t-1 ln(2t - 1) t² - 1 r(t) = (VI- t-1 2 COS πt rt), (t (t = 1):…

A: Let us solve above problem

Q: tangent plane at the point (-2,1 – 3)

A: Equation of the tangent plane = dfdxx-x0+dfdyy-y0+dfdzz-z0 Let f=x24+y2+z29-3 (given)

Q: Compute the directional derivative of f(r. y) = = at the point (1, 1) in a irection normmal to the…

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Q: V = 6y – 16x + 9z + V¢ Find the curl of vector V in Cartesian coordinates, o is Continuously…

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Q: Suppose F(x, y) = x² + y²j and C is the line segment segment from point P = (-3,-4) to Q = (-3, 1).…

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Q: (a) Find a vector function that describes the curve of the intersection of the cylinder z²+²=4 and…

A: We will use trigonometric identity to find parametric equation of curves and combined them to find…

Q: 1 (d) Last week, Amy walked a median of 7.5 kilometres each day. 1 kilometre = 0.6 miles The table…

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Q: Find the directional derivative of f(x, y, z) = x²y²z + 4xz at (-1,2,1) in the direction 2i + j +…

A: We have to find the directional derivative of f(x, y, z)=x2y2z+4xz at (-1, 2, 1) in the direction…

Q: Let P be the point (2, -1, 2) and Ø,: x² + y² – z² = 10. (i) Find the unit normal vector to the…

A: (c) The given surface is ∅1: x2+y2-z2=10 and the point P: 2,-1,2. (i) The unit normal vector to the…