Find T, N, and k for the space curves in Exercises 9–14. 9. r(t) = (3 sin t)i + (3 cos t)j + 4tk 10. r(t) = (cos t + t sin t)i + (sin t - t cos t)j + 3k 11. r(t) = (et cos t)i + (et sin t)j + 2k 12. r(t) = (6 sin 2t)i + (6 cos 2t)j + 5tk 13. r(t) = (t3/3)i + (t2/2)j, t> 0 14. r(t) = (cos3 t)i + (sin3 t)j, 0 <t< pai/2

Q: Sketch the space curve and find its length over the given interval r(t) = ⟨8 cos t, 8 sin t, t⟩,…

A: Given Finding the derivative with respect to t:

Q: Let F(x, y) = x2 + x² + y2J- (a) Show that F. dr +| F. dr if C and C2 are the semicircular paths…

A: The function given to us is  F(x,y)=yx2+y2i-xx2+y2j

Q: Find the linearizations L(x, y, z) of the functions in Exercise atthe given points.ƒ(x, y, z) =…

A: To findLinearizations of f(x,y,z)=tan-1xyz at (1,0,0) , (1,1,0) and (1,1,1)

Q: Show the function f(z) = z (z) is conformal at 0 and f'(0) = 0. Does this function violate our…

A:

Q: Use Green's Theorem to compute y dx+(x³+3xy²) dy where C is the closed curve around the triangle…

A:

Q: Exercise (2): Calculate S. xydx + 3y*dy where C is the curve in xy plane, x = y² from (0,0). to…

A: We have to find the line integral

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: Find T, N, and k for the space curves r(t) = (6 sin 2t)i + (6 cos 2t)j + 5tk

A:

Q: 7. Given the space curve er(t) = (4 sin t ,4 cos t , 10t), Find T, N, B, and t (torsion)

A:

Q: sketch the solid whose spherical coordi- nates (p, 4, 0) satisfy the given inequalities. 0<p<2/cos…

A:

Q: Sketch the space curve. Vector-Valued Function r(t) = (2 sint, 7t, 2 cos t) 1.0 0.5 0.0 0.6 -0.5]…

A:

Q: 15. What is the correct piecewise parametrization of a path C created by starting from C₁ of…

A: Introduction: The parametrization is a very significant part that is related to vector calculus.…

Q: 2. Let L be the line tangent to the space curve represented by r(t) = (t² – In(t))i+(e²-t – t)j+…

A:

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: 1 4. Let f(z)=: Show that J(z))dz = 0 , where Cis a simple closed contour not passing throughout…

A: Given information : f(z) = 1z2    and Cis a simple closed contour not passing through origin.

Q: Let C be the space curve defined by Ř(t) = (2cos t,2 sin t, t). O Find the moving trihedral of C at…

A:

Q: Show that the general solution to the two-dimensional Laplace's Equation. et = 0 is (Acosh(tx)…

A: We have to show that the general solution to the two-dimensional Laplace's Equation…

Q: Find a parametrization for the line segment joining points (0, 2) and (4, 0) using the angle 0 in…

A: Modify the given figure as shown in Figure 1.

Q: (b) Give an example of a parametrisation of the unit circle, C = {(x, y) E R² | x² + y² = 1}, that…

A: The parameterization of a circle  that generates the clockwise orientation of C is given by: x=r…

Q: Sketch the space curve r(t) = a cos ti + a sin tj + btk and find its length over the given interval…

A: Given : Space curve equation rt = acosti^ + asintj^ + btk^  ,   and  t ∈ 0,2π  ,a and b are contants…

Q: 4. (Section 17.8) Use the Divergence Theorem to calculate [[F F-dS, where F = y + ryj – zk and S is…

A: To use the divergence theorem to calculate ∫∫SF⋅dS.

Q: Exercise 1.3.2 Give a paramatrization for the curve curve y = x³ + ()x-¹, then deduce the length of…

A: Given curve is  y=16x3+12x-1

Q: Let r(t) = (sin(t), cos(t), 5 sin(t) + 4 cos(2t)). Find the projection of r(t) onto the xz-plane for…

A:

Q: 15. What is the correct piecewise parametrization of a path C created by starting from C₁ of…

A: Given Path C is C1:  y=x+3,     0≤x≤4C2:  y=x2,         y=7 to y=0C3:  line connecting C1 and C2…

Q: 3. Sketch the space curve, r(t) = 8 cos ti +8 sin t j+tk , and find its length over the interval []…

A: Given: r(t)=8cost+8sint+tk    [0,pi/2]

Q: Identify the symmetries of the curves in Exercises 1–12. Then sketch the curves in the xy-plane. 1.…

A:

Q: Find the lengths of the curves in Exercises 25–27. 25. x = cos t, y = t + sin t, 0 &lt;=t&lt;= pai…

A: The given curve is

Q: 7. Given the space curve r(t) = (4 sin t,4 cos t, 10t), Find T, N, B, and T(torsion)

A:

Q: To illustrate that the length of a smooth space curve does not depend on the parameterization used…

A:

Q: Use Green's Theorem to evaluate the line integra Icon (y − x)dx + (2x − y)dy - - for the given path.…

A:

Q: Find a parametrization for the portion of the sphere x2 +y2 + z2 - 16 between the planes z =- 2/2…

A: we have been given a sphere x2+y2+z2=16 between the planes z=-22 and z=2

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…

A:

Q: Evaluate R CF. dr with F(x, y, z)=(x2 + y)i – (3xz)j + (z – 4y) k, with the path C as the line…

A:

Q: Find a piecewise smooth parametrization of the path C. y 5 (5, 4) 4 3 1 2 3 4 5 (5 - t)i + (4 - t…

A:

Q: Find T, N, and B for the given space curve. r(1)= (6+t)i + (2 + In (sec t))j-3k, - T/2 <t < T/2 O A.…

A: Find r'(t) by differentiating r(t) with respect to t. r(t)=6+ti+2+lnsec tj-3kr'(t)=i+sec t·tan tsec…

Q: Let Ci be the rectangle with vertices (0,0), (2,0), (0,1), and (2,1) oriented counterclockwise, and…

A: Let C₁ be the rectangle with vertices (0,0). (2,0), (0, 1), and (2,1) oriented counterclockwise, and…

Q: To illustrate that the length of a smooth space curve does not depend on the parameterization used…

A: Since you have posted multiple questions but according to guidelines we will solve the first…

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: Sketch the space curve r(t) = ⟨4t, −cos t, sin t⟩ and find its length over the given interval [0,…

A: Consider the given space curve, rt=4t,-cost,sint use the graphing utility to graph the curve,

Q: find ⌠⌠F*dS if F(x,y,z)=2xi+2yj+2zk and S is the paraboloid z=8-x^2-y^2 lying above the rectangle…

A:

Q: Let r(t)=⟨sin(t),cos(t),9sin(t)+3cos(2t)⟩. Find the projection of r(t)r(t) onto the xz-plane for…

A: Set y=0 as the answer is asked for xz plane.

Q: Use Stoke’s theorem to evaluate ꬹϲ F.dr, if F(x,y,z)=(-xy,)I –(xz)j+(2yz)k and C is the closed…

A: The objective is to use the Stokes' theorem to evaluate the integral ∫CF·dR where…

Q: 1. Use a change of variables to evaluate [[ y- dA where R is the trapezoid with vertices (1,0,…

A:

Q: Sketch the space curve. Function r(t) = (5 sin t, 3t, 5 cos t) 1.0 0.5 0.0 0.0 -0.5 -1.0 -5 N 0.5…

A: Please see the below picture for detailed solution.

Q: Sketch the space curve. Vector-Valued Function r(t) = (4 sin t, 7t, 4 cos t) __4_2 1.0 0.5 0.0 0.0…

A: Given data÷   rt=&lt;4sint,7t,4cost&gt;   For curve option A is correct one.

Q: To illustrate that the length of a smooth space curve does not depend on the parameterization used…

A: To find - To illustrate that the length of a smooth space curve does not depend on the…

Q: Sketch the plane curve. r(t) = 2ti-tj, [0, 2] y -1 y 2 1 Find its length over the given interval. 2…

A:

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.