.3 Evaluate || F · dS, where F = < y, – x, z x, z° > and S is the helicoid with vector equation r(u, v) = < u cos v, u sin v, v > 0

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: Which of the following is the directional derivative of f(r, y) = at the point (0,0) in the…

A:

Q: If an object follows the path of the two-dimensional vector-valued function p(t)=(16−cos(t),…

A: Given p(t)=(16−cos(t), 0.6πt−0.15t2−3)

Q: The monkey saddle z = x³ - 3xy² can be intersected with the cylinder x² + y² = 1 to produce the…

A: Given monkey saddle is z=x3-3xy2 and the cylinder is x2+y2=1. The given curve is the intersection of…

Q: Show that this curve y represents the vector (1,0) as v = 1- +0- in the sense that, for an arbitrary…

A: Given that γ(t)=(t,t2)in R2, t∈[-1,1] The objective is to show that curve represents the vector…

Q: ix of F across the surface defined by R(u, v) = (2u, 3v E [0, 1].

A: We have to find the flux according to the given information

Q: (d) Use Stokes' theorem to evaluate S. F.dr where = z²î + y²j + xk and C is the triangle with…

A:

Q: Evaluate the line integral ∫cF⋅dr, where F(x,y,z)=−3sinxi−2cosyj−xzk and C is given by the vector…

A:

Q: Consider the following vector function. r(t) = (t, t?, 4) (a) Find the unit tangent and unit normal…

A: Given vector r(t)=t , t2 , 4

Q: Evaluate fc F • dr, where F = (z, y2, x) and C is parametrized (in the positive direction) by r(t) =…

A: F=z,y2,x and C is the parametrized by r(t)=t+1,et,t2 for 0&lt;t&lt;2. We have to evaluate ∫CF·dr.

Q: Find an equation of the tangent plane to the surface represented by the vector-valued function at…

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Q: 3. Let the curve C be traced by the vector-valued function R(t) = cos(zt) î + at j+ sin(rt) k. %3D…

A:

Q: Let C be the curve of intersection of the paraboloid x? + y² = z and the cylinder + (y – 1)² = 1. 4…

A: We will find equation of common curve and parametrize it to find vector equation of C and the use…

Q: 18. Determine the values oft between 0 and 27 such that the tangent vec- tor to the cycloid r(t) =…

A:

Q: Prove that C is tangent to S at (0, -1, 1), considering that C has a vector-valued function 2 i+ 4 4…

A: Here, we are given the following equation of surface S : x3+y3+z3=xyz. Let us take f(x, y,…

Q: ii) Evaluate the vector line integral of F, F. ds, where y is an oriented curve defined as follows:…

A:

Q: Given the function f (x, y) = 2?y - tan-(xy) Itan-(ay) %3D Find the vector normal to the level curve…

A: Find the normal vector to the level curve at a point

Q: 6. Given the space curve r(t) = (t, 2et, e2t), find the tangential and normal components of…

A:

Q: Find the work done by force F(x, y) = 4yi + 7xj + (x + y)k in moving an object along curve r(t) =…

A: To find the workdone along the given path

Q: H=pz coup ap te'sing a4 tp'ar At point (1, ,0)

A:

Q: 2. Consider the curve traced out by the vector valued function r(t) = (sin t cos t, cos? t, sin t)…

A: given that r(t)=(sintcost,cos2t,sint) we need to show that the curve lines on the surface of the…

Q: Consider the curve described by the following vector function. r(t) = (90 e - cos t) i+ (90 e -* sin…

A:

Q: 2. Consider the following vector function: r(t) = = a cos wti + a sin wtj + bwtk, where a, b and w…

A: We need to find radius of curvature.

Q: 2. Find the unit tangent vector T(t), principal unit normal vector N(f), and curvature x, for the…

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

Q: ). Determine the set Tp(Zf) of all tangent vectors to Zf at p. Is dim Tp(S) m ker d f.?

A: Sol

Q: need some help with this

A:

Q: Which of the following is NOT true about the vector function r = r(t) where |r |= constant

A: yes, given statement is true. for a given vector function r = r(t)r = constant always true

Q: 14) Consider the curve C in R3 whose parametrization is given by r(t) = ( + 1,t³ – t, vE) con te Let…

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Q: Find the unit tangent vector r(t) = t2i + tj + 4/3k and a set o f parametric equations P(1, 1, 4/3)…

A:

Q: 13. Given the space curve r(t) = (e' cos t , e' sin t , e“), find the tangential and normal…

A:

Q: Which of the following vector functions represents the curve of intersection of the cylinder x2 +y?…

A: Given x2+y2=4 and plane x + z =3

Q: Given r(t) be the vector function that describes the curve of intersection between the cylinder…

A: First, parametrize the cylinder equation to get x(t) and y(t) and then find find z(t) from the…

Q: Show that the vector-valued function r(t) = ti + 2t cos tj + 2t sin tk lies on the cone 4x2 = y2 +…

A: Given: Vector-valued function: r(t)=ti^+2tcostj^+2tsintk^ Cone: 4x2=y2+z2

Q: Consider the curve r(t) = (t, 3 cos t, 3 sin t) for t 2 0. Is the integral o r(t)dt a number or a…

A: Given :   r(t) = ( t , 3cost , 3 sint )

Q: Find a tangent vector at t= In 2 for the following parameterized curve. r(t) = 5 e'i+e -2tj+6 e 2k…

A: Explanation of the answer is as follows

Q: Find the work done by force F(x, y) = 2yi + 3xj + (x + y)k in moving an object along curve r(t) =…

A:

Q: 7. The magnitude of the normal vector t, xt for a sphere of radius a in parametric form where u = p…

A: Radius of the sphere is a. It is given that,u=ϕv=θ In parametric form, the sphere can be represented…

Q: Let r(t ) = ti + cost j + t? k be a space curve. Which of the following vectors is parallel to the…

A:

Q: 3- Calculate the curls of the following vector functions: (a) V = s(2 +sin Ø) s+ s sinØ cosØ Ø+ 3z…

A: V→=s2+sin2∅∅^+ s sin∅cos ∅ ∅ ^+3zz^ curl V→=∇×V→=s^∅^z^∂∂s∂∂∅∂∂zs(2+sin2s)s…

Q: Find the unit tangent vector at the indicated point of the vector function r(t) = e3t cos fi+ et sin…

A: We use the following formula to find the unit tangent vector T(t)=V(t)/|V(t)|

Q: Let r (t) = ti + e2t j + t³ k be a space curve. Which of the following vectors is perpendicular to…

A:

Q: Suppose r(t) is a vector function in R3 Show that if r(t)is perpendicular to the tangent vector…

A: Suppose that the position vector r→t be, r→t=ft,gt,ht…

Q: Which of the following is the directional derivative of f(r, y) at the point (0,0) in the direction…

A: Formula to find the directional derivatives helps to find the required directional derivative in the…

Q: ; Consider the vector function r(t) = ti + t²j+t°k. Compute the curvature of r at time t = 2.

A: Hello. Since you have posted multiple questions and not specified which question needs to be solved,…

Q: 3. Consider feenction fi CR?> R? given lby (a) Grodient vector ♡ f Qt poiat Cuo) es D (6) tangent…

A:

Q: (b) Use the corresponding formula to evaluate the line integral F. ds where F(x, y, z) = (xz, -yz,…

A:

Q: 9. Evaluate the line integral ſo F · dr, where F(x, y) = (ry, 3y?), and C is is given by the vector…

A: We have to integrate F • dr over the curve C given by the vector function r(t) = ( t4, t3 ) for t in…

Q: 1. Find the divergence of vector O = p?sinØ ap + z aø + pz aż and evaluate at these points sec O…

A:

Q: 1. Let C be the curve of intersection of the paraboloid x2 + y? = z and the cylinder 4 - + (y – 1)²…

A: Given: The paraboloid equation is x2+y2=z and the cylinder x24+y-12=1 a) Find the vector-valued…