1. Show that the curve a(t) = (t,t² + 1, t – 1) is regular.

Q: 3) Consider the curve C in R³ given by: r(t) = (t + In t, t – In t, 4t'/²) con 1<t< 2 An expression…

A: Formula: If r(t)=x(t), y(t), z(t), a≤t≤b is parametrization of a curve C then the line integral…

Q: 4) 3 Find the length of the curve r(t) =t i+t°j+2r°k if 0<t<1.

A: r(t)=f(t)i+g(t)j+h(t)k  ;a≤t≤b Curve length =∫ab√(f′(t))^2+(g′(t))^2+(h′(t))^2dt.

Q: Evaluate Se (5xy + x² + y²) dx + (x² - y)dy by using Green's theorem, where C is a closed curve that…

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Q: (b) Verify that Green's Theorem is true for S,3(y² – x) dx + (x² – y)dy given that the curve C is…

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Q: Consider the two parametrized paths: r(t) = (t² + 7, t + 1, 25t-1), s(t) = (8t, 2t – 2, t² – 8) What…

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Q: Find a parameterization for the curve y = 7 – 2x that passes through the point (0,7, 3) when t = 2…

A: To find the parameterization of the curve

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 the work done by F in moving a particle once counterclockwise around the given curve. F = 2xy3i…

A: Consider F = 2xy3i + 4x2y2j We have M = 2xy3 and N = 4x2y2

Q: find the work done by F in moving a particle once counterclockwise around the given curve. F = (4x -…

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Q: 2 32 |for 1st<3. length of the curve c(t) =| In t,-t"

A:

Q: Use Green's theorem to evaluate S. (5xy + x² + y²) dx + (x² – y)dy where C is a closed curve that is…

A:

Q: find a parametrization c(t) of the curve satisfying the given condition.  y = x 2, c(0) = (3, 9)

A: The given function is y=x2 and c(0) =(3,9) Let x(0) =3 and y(0)= 9 Take x =t +x(0) Thus, x = t+3

Q: Sketch the graph of 7(t) = sin(t)i – 2cos(t)j–tk in 3-space; then find and draw the tangent vector…

A: Given, r→t=sinti^-2costj^-tk^

Q: 30. (2x + y²) dx + (2xy + 3y) dy C C: Any simple closed curve in the plane for which Green's Theo-…

A: We know that Green's theorem

Q: At what points does the curve  r(t) = ti + (5t − t2)k  intersect the paraboloid  z = x2 + y2

A: We have to find the intersection point

Q: Sketch the plane curve r(t) = t3i + t2j and find its length over the given interval [0, 1] .

A:

Q: find the decomposition of a(t) into tangential and normal components at the point indicated, as in…

A: Consider the given information rt=eti+1-tj ,  t=0   Find the first derivative of rt…

Q: s Compute yz dz + zzdy + zydz along the curve (t, t2, t³), 0 st<1

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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: Use Green's theorem to evaluate (2y dx-x dy) around the closed curve C. 27 (2,8) 8. 4. C. -2 (1,-1)

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Q: find the decomposition of a(t) into tangential and normal components at the point indicated, as in…

A: Given: r(t)=(t,cost,sint)t=π2

Q: 2. Let F(t) = (e' cos t) î+ (e' sin t)j+ek. a. Find symmetric equations of the line tangent to the…

A: Given:  F→t = et costi^ +et sint j^ +et k^  To Find:  Symmetric equations of the line tangent to the…

Q: Compute I = (2x²132²y+ 4xy)ax +(x'+2x²-y') dy+zdz along The curve C C4U Cz UG given in the figure…

A: Given To compute I=∫C2x3+3x2y+4xy.dx+x3+2x2-y2dy+zdz along the curve C=C1∪C2∪C3 given in the figure.

Q: 19. Evaluate Scx ds, where C is a. the straight-line segment x = t, y = t/2, from (0, 0) to (4, 2).…

A: Part (a)

Q: Find the work done by F = xyi+yj - yzk over the curve r(t) = ti + t²j + tk, 0≤t≤1

A:

Q: 6. (a) Show that F = (3a²y+2, x3 + 3) is conservative. (b) Evaluate Sc(3x²y +2)dx + (x³ + 3)dy where…

A: According to the problem, we have

Q: 2. Find the work done by F in moving a particle once counterclockwise around the given curve C F =…

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Q: What is the slope at t = −2 ?

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Q: 5. Let C be the curve parameterized by r(t) = (t,-t,t²) Evaluate y ds.

A:

Q: 1. Sketch each curve in the z-plane, and sketch its image under w = 2². (a) |z1|=1 (b) x = 1 (e) y²…

A: We are authorized to answer three subparts at a time since you have not mentioned which part you are…

Q: Evaluate x*dx C + xy dy, where C is the triangular curve consisting of the line segments from (0, 0)…

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Q: find a parametrization c(t) of the curve satisfying the given condition. y = x2, c(0) = (3, 9)

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Q: 1. Find the exact length of the curve. (a) y² = 4(x+ 4)°, 00 (b) y = In z, 1<I<2

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Q: Determine the torsion at t = 0 of a curve a : [−1, 1] → R3, a(t) = (2t, t^2, t^3/3). Moreover,

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Q: 5. Compute the lenght of the curve u = t, v=0 from t = 0 to t = 1 using the first fundamental form.

A: Suppose, a continuous function is provided over a certain interval. The arc length is the length…

Q: 7. Find the exact length of the curve: x (t) = 2t, y(t) = 2t, 0 ≤ t ≤ 1

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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…

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Q: Find the length of the curve r(t) = (√2t)i + (√2t)j + (1 - t2)k from (0, 0, 1) to (√2, √2, 0)

A: given, r(t)=(2t)i +2tj+1-t2kwe have to find the length of the  given curve from (0,0,1) to 2,2,0

Q: 1. Compute (1/zy, 1/(z + y))- dr along the curve (t, t2). 1<ts4

A:

Q: 1. Find the tangent vector to the curve 6t r(t) = [t³ – t, t+1 , (2t + 1)*] at t = 1.

A: As per our guidelines we are supposed to answer only one question. If you want solution of any…

Q: determine the parametric coordinate of the critical point of the curve x=t^2+3t=2 and y=t^2-1.

A: Curve will have critical point if it have vertical tangent or horizontal tangent . Slope of…

Q: Find the family of curves orthogonal to the family 4y2 + 3x2 = k when x > 0 and y > 0. (A) 3y2 – 4x2…

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Q: ind the orthogonal trajectory of the curve y? = ax²(1- cx); with a held fixed.

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Q: Show that the graph of ř = (t sin t, t cos t, t²) lies on the paraboloid z = x? + y?.

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Q: 2. If F(x, y, z) = y²i+(2xy+e3=)j+3ye³=k. Evaluate F·Î ds, where C is the curve represented by r(t)…

A: Given F→x,y,z=y2i^+2xy+e3zj^+3ye3zk^ and the curve is C given by r^t=cost,t,sint,  0≤t≤π2. Find…

Q: Evaluate | 9+x* dx+xy*dy along the positively oriented curve C where the C is the boundary of the…

A:

Q: Find the diffrential equation of for each of the curves determined by the condition that at each…

A: length of subtangent =ydydx

Q: (1) Find the curl of the curve F=(x*y.y*z.2x) (0.0. y+1) (-x'.2'.y+1) b) d)

A: We have to find the curl of the curve F=(x2y,y2z,z2x) curl is defined as the cross product of ∇ with…

Q: 1. Find the tangent vector to the curve 6t r(t) = [t° – t, (2t + 1)*] at t = 1.

A: