a) +++ where C is the helix r(t) = cos(t) i + sin(t)j +t k, Osts2T

Q: · Find fF · dĩ for F = -yi – 4x j+4zk if C is the helix æ = cos t, y = sin t, z = t, for 0 < t < 47

A: Given F→=-yi→-4xj→+4zk→

Q: A particle moves in the xY plane so that at any time 0 <t < 4, x = 2 cos t and y =t² + 3 sin t + 5…

A:

Q: 4. Evaluate 2.x cos y dx – x sin y dy along the elliptical path + y? = 1 from (2,0) to (0, 1)…

A:

Q: Consider the helix r(t) = (cos(4t), sin(4t), –2t). Compute, at t

A: Compute unit binormal and curvature at t=pi/6

Q: C: r(1) = (1 – cos t)i + sin tj Evaluate F. dr, where F(x, y) = (y + 1)i + (3xy² + 1)j 1- and C, is…

A:

Q: Find the length of one turn of the helix given by…

A: Given: The parametric equation of the curve: .

Q: 32. At what points does the helix r(t) = (sin t, cos t, t) intersect the sphere x? + y² + z² = 5?

A:

Q: 4- if U(x,y) = x2 + 2y and x(r, t) = r sint, y(r,t) = sin?t, au find 1- ar au and 2- at au 1- ar ди…

A: Given that,   U(x,y) = x2+2y   and x(r,t) = rsint,     y(r,t) =sin2t we have to find ∂u∂r,    ∂u∂t…

Q: If z= r +y, I= e' cos(s) and y= e' sin(s), then %3D 8z as at (t, s) = (1,0) is O-2e O 4e O 2e? 2e

A:

Q: 4e cos z Integrate (z counterclockwise around the circle |2| = 3. :- )* 4e? cos z dz (z

A:

Q: Find S, F· dř tor F = -5yỉ + 4x} – 4zk if C is the helix x = cos t, y = sin t, z = t, for 0 < t <…

A:

Q: O(y-+ rycota)dr + zdy O (coszcosy– cotz)dr sinrsinydy 0 O ydr + (3z -g')dy = 0

A: A differential equation is called linear if 1) every dependent variable and derivative involved…

Q: 4. Calculate leme/) at (z,y) = (a, 4). sin(z/g) ey2/2 A ev2/2 B 4

A: Here we have to calculate ∂∂xesin(x/y)=? at (x,y)=(π,4)

Q: et u = e Sin P= sin sc Lind su using chain Rule

A:

Q: šiven r,(t)=sin5ti +cos 5tj+\t+2k and r,(t)=5ti+2k , calculate "[r, (0) •r, (0)]. dt

A:

Q: Consider a particle moving along a path C given by r(t) = v3ti+ = V3ti sin (t)j + cos (t)k Find the…

A:

Q: 4.1 z + cos(72) -dz, c z(z² +1) with C : |z – i| =.

A:

Q: Find the length of the hypocycloid defined by the equations x = a cos° (0), y = a sin° (0), 0 < 0 <…

A: Here we have to find the length of the hypocycloid defined by the equation x=acos3(θ) , y=asin3(θ)

Q: 7. (~Ch16 Review #6) Compute y3 dx + x² dy where C is the arc of the parabola x = 1- y² from (0, –…

A: In this question we need to find the given line integral .

Q: Consider the curve C parametrized byx = −1 + 6 sin t and y =(1/2)− 6 cos t for −π ≤ t ≤ 7π. What is…

A: First of all, consider the given parametrized equation of curve C. Rearrange the equations to write…

Q: Let f(r, y, 2) = 2ry + 2. Evaluate f(1, y, 2) dz where C is the portion of the helix r(t) = cos ti+…

A:

Q: 3. ® Find the genelal Solution of the differentral egation (D'+ 25) y = cscX Sinx t Sin 5X cos 3X

A:

Q: | Find fe F dr tor FF = -5yi+3x j+4z k if C is the helix x = cos t, y = sin t, z = t, for 0 <tS 4x.…

A:

Q: au %3D ax at u(x, 0) = 0 ди (x, 0) = 0 at u(0, t) = sin? t u(1, г) — 0,

A: Given,           ∂2u∂t2-∂2u∂x2=0                       for 0&lt;x&lt;1, t&gt;0      ...(1)u(x,…

Q: 4.1 z+ cos(T2) dz, z(z² + 1) with C: |z – i| =.

A:

Q: Find the length of the curve. r(t) = cos(4t) i + sin(4t) j + 4ln(cos(t)) k,  0 ≤ t ≤ ?/4

A:

Q: Find D (x, y, z) where f(x, y, z) = cos(z²)sin(y²) In the direction of v = (4, 15, 18). %3D Possible…

A: Introduction: The formula for the directional derivative of the function f(x,y,z) in the direction…

Q: (a) Evaluate where c is z = 1 %3D c cos(Tz)

A:

Q: 2. Find the length of the asteroid x = cos't and y=sin't , 0<t<2n. x = cos' t y = sin³ t

A:

Q: U + ru, =x u(z,0) = sin(87x) cos(90r). IE R, t> 0,

A: Given ut+xux=x2 By Lagrange's method , we get dt1=dxx=dux2 By equation (1) and equation (2) , we get…

Q: Use Green's Theorem to evaluate | (y? – tan-1 x²)dx + (3xy + 4x – In (y² + 4))dy where C is the…

A:

Q: Find the length of one turn of the helix given by r(t) == cos(t)i + sin(t)j + 1 1 99 25 -tk.

A:

Q: dz and du dz Use an appropriate form of the chain rule to find dv z = =; x = 16 cos u, y = 13 sin v…

A: In this question we have to find differentiation with respect to u and v.

Q: 1. Find the length of the curve r(t) = (sin 2t, cos 2t) from t 0 to t= 2n. CoS %3D

A: Since you have posted multiple question in this question which are not interlinked so i have solved…

Q: Example: Find the torsion of the r= (cos t i + sin tj + tk) 1

A:

Q: Q:Ifx= sin 6, y = 0,y = cos pe, prove that (1-x )y,-xy,+p°y = 0, %3D dy where y, dy and y1 d

A: Here given that x=sinθ, y=cospθ Taking derivative of both x &amp; y with respect to θ, we get…

Q: Find the length of the arc from 0 = 0 to 0 = 2n for the cardioid r = f (0) = 2 – 2 cos 0.

A: Solution if r=fθ  has a continuous first derivativefor α≤θ≤β then length of polar…

Q: S. Re z dZ where c is the shorfest tra Jectory from Co,0) #i o (Iti) 2 Re z dZ Where cis a di sh…

A: (a). Consider the integral ∫CRezdz, where C is the shortest trajectory from 0+i0 to 1+i. Let z=x+iy,…

Q: 6) Find the equation of the osculating plane of the helix x = sin 3t, y = 2t, at the point (-1,л,0).…

A: Given :x=sin3ty=2tz=cos3tpoint -1, π, 0

Q: - A duck is flying the helical path r cos(t), y = sin(t), and z = t with t in seconds. TE emperature…

A:

Q: - Find the aera under the curve y = cosx Form x = 0 to x = -fx sec² (ax² )dx a#0

A:

Q: Find the length of the curve. r(t) = cos(3t) i + sin(3t)j + 3 In(cos(t)) k, 0 <t< n/4 %3D

A: Find the length of the curve:…

Q: 2- Find the solution on the exterior of a sphere of radius a 1 or (r F) + Sin e 0 (sin 0) = 3u, u…

A: Given: ∂∂rr2∂u∂r+1sinθ∂∂θsinθ∂u∂θ=3u⇒r2∂2u∂r2+2r∂u∂r+1sinθsinθ∂2u∂θ2+cosθ∂u∂θ=3u…

Q: dz az If z = xyeX/Y, x = r cos(0), and y = r sin(0), find and ar when r = 2 and 0 = %3D az ar (r, 0)…

A:

Q: Find Sc F dr for F = 2yi + 5x j+ 3zk if C is the helix æ = cos t, y = sin t, z = t, for 0 <t< 4T.…

A:

Q: 7e cos z Integrate counterclockwise around the circle |z| = 8. (z – 3)² - 7e cos z dz 2 c (z – 5)

A: By using  Cauchy integral formula we can solve this question.

Q: 4. r(t) = (cos t)i + (sin t)j + (sin 2t)k, t, 2 %3D

A:

Q: 11 z- CCW -dz, z2 - z de. c:|2- = sec(u), <ý<.cv Question 11) Find I =

A:

Q: 7. Let F = (2r sin(y)e", x² cos(y)e", a² sin(y)e*). %3D Find along the path c:7 = (sin(t), t,…

A: Given F→=2xsinyez,x2cosyez,x2sinyez. Find ∫CF→·dr→ along the path c:r→=sint,t,cost,-π2≤t≤π2. From…

Q: 74. A particle moves in the xy-plane along the elliptical path x = 2 cos t, y = sin t, where x and y…

A:

Estimation

A new framework of data analysis where the new statistical methods are used for interpreting the results and analyzing the data is known as estimation in statistics.