1. Which of the following(s) is/are applicable to solve | sin(2y)dr – y cos(r)dy over an arc from (-V6, v3) to (v6, V3)?(Select all that apply!) A) Line Parametrization B) Fundamental Theorem of Line Integral C) Green's Theorem D) Reversing Orders

Q: 14) *Find the arclength of f (x)=x2 on [0,2]. %3D

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)…

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Q: 2.2 Prove that Te -dz < 3 ez z2 +7+1 where C is the arc of the circle |2| = 3 from z = 3i to z = -3.

A:

Q: 77. Prove, in two ways, that the arc length of a linear function f(x) = mx + c on an interval [u, b]…

A: f(x)=mx+c on an interval [a,b] Note: we prove part a) only ,in part b) theorem 6.7 not given

Q: 4. Sketch the curve of r2 = cos 20 and find the area that encloses it. Show your complete %3D…

A: Converting polar coordinates to rectangular coordinates  Put r=x2+y2,  θ=tan-1yx in r2=cos2θ…

Q: 1. Use Green's theorem in order to compute the line integral $ (3cos + 6y?) dx + (sin(5") + 16x³) dy…

A: Here the given integral is                               ∮C3cosx+6y2dx + sin5y+16x3dy

Q: 6. (The fractional residue theorem) Show that sin(ar) L r(r² + 1) tz = 5(1 – e~“), a>0. iaz [Hint:…

A:

Q: Prove that fe, r*dx + rydy = 0 where Co is the line segments from (0, 1) to (0,0) by using…

A:

Q: Which of the following(s) is/are applicable to solve 3e dx – 6xe dy over triangle whose vertices are…

A: To find method applicable to solve ∫C3e-ydx-6xe-ydy over triangle whose vertices are (0,0),(4,-1)…

Q: 13.9. Let be the arc of the circle 2 = 2 from z = 2 to z = 2i that lies in the first quadrant.…

A:

Q: 3. For what value of K the velocity field in plane polar coordinates given by v, = tion of…

A: 3. The given values are,vr=K36r2-1cosθ and vθ=k36r2+1sinθ

Q: 1 Evaluate f - dz over the unit circle in counter-clockwise direction using the parametriza- tions

A: To evaluate, ∮1zdz over the unit circle counter-clockwise

Q: Q3 Point out the singularity and is ty pe of follo wing fune tions 22 () d) Sin t2 24

A: Singularity

Q: What is the order of the pole of f(z) = 1 (2 cos z −2+z²)² at z=0? [HINT: Work with 1/f(z).]

A:

Q: Show that the spherical harmonics 1 3 sin 0 e-iº, 2V 27 Y1,+1 and 3 1 Y1,0 2V V cos 0, %3D | are…

A: Given: The given harmonics are Y1,+1=1232πsinθ e-iϕ and Y1,0=123πcosθ  are orthogonal over the…

Q: Calculate the volue of the de integral 23 - 1 2 F soro fallowing c a) Cil2+1|= 1 Curves. 6)…

A: Since you have posted a question with multiple sub-parts, we will solve first three subparts for…

Q: Jse Green's theorem to evaluate the ne integral (2tan x _cos 2y³) dx + (7⁰s y + 2x (πº where is the…

A:

Q: (1) (a) Show that the line integral tan(y) dr +x sec (y) dy is independent of path, where C is any…

A:

Q: (a) Set up an integral for the length of the arc of the hyperbola xy = 1 from the point (1,1)  to…

A: Part (a)

Q: Q1- The divergence of the function f-(rs cos 0)e, in spherical coordinates is......

A:

Q: Q1- The divergence of the function f= (r cos 0)e, in spherical coordinates is..........

A:

Q: Use Green's Theorem to evaluate F. · dr, where F = (Va + 4y, 2x + and C consists of the arc of the…

A:

Q: 35)16 pointsi se t up, but Po not , the definite integra! that computes the are enclose ol intsi Se…

A: The interval –π/4≤θ≤π/4, could be divided into  –π/4≤θ≤0 and 0≤θ≤π/4

Q: The value of cosh(3z+1)dz, where C is the line segment from z=1 C to z= 2i. O a. (-sinh(6i+1)+sinh…

A: We have to find the value of integral,  I = ∮Ccosh(3z+1)dz  ;  where C is the line segment from z=1…

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: 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: A- Find all values of z such that sinh(z) = B- Are |z, Img(z) analytic? Give reason. C- Choose the…

A: We have to solve the given questions

Q: Which of the following is an arc-length parametrization of the line segment from (2,1) to (1,3)?…

A: We will find out the required value.

Q: 2. What is the arc length of the polar curve r = 1- sin 0 for 0 <0< <5? 2 (a) 2(v2 – 1) 2 - V2 2/2…

A:

Q: Use Green's Theorem to evaluate F. dr, where F C consists of the arc of the curve y 1x x2 - from…

A:

Q: If F(x, y) = (6xy - 1,3x2 + 2y) and C is the circular arc along x² + y? = 2 from (1, 1) to (-1, 1),…

A: Given vector field is  F(x,y)=6xy-1,3x2+2yF(x,y)=(6xy-1)i+(3x2+2y)j c is the circular arc along…

Q: - Prove that the contour integral – z)dz = -12i where C is the triangle with points 0, 3i and -4…

A:

Q: 9.40. Show that the 0-parameter curves on the surface x = (r cos 0)e¡ + (r sin 0)eg + f(0)eg are…

A: We have to show that the θ-parameter curves on the surface x are parallel.

Q: Find the arc length of f(x) = from x = 0 to x = V3. 2 - 1, 13 a) О O + In(2 – v3)]I 2 b) OIV3 + In(2…

A: Length of Arc =∫x=ax=b1+dydx2dx

Q: Evaluate the line integral S¢F dr where F(x, y) = xsin(y)i + yj and C is the arc of the parabola…

A:

Q: 9. Using the fact that lim | sin z e*dz = 0 , where C, = Re" , 0<oST, and that sin z has simple pole…

A:

Q: 8.) Find work done by F = (2x³y* + x, 2x*y³ + y) along the path r(t) = (t cos(at) – 1, sin (t))…

A: Since we have given that F = 2 x3 y4 +x, 2 x4 y3 +y   and rt=t cosπt-1, sinπ2t    drt = cosπt -t…

Q: Use Green's Theorem to evaluate the integral Sc(5 sin y – 3y, 2 + 5x cos y) · dĩ (6) where C is the…

A: (6) Use Green's Theorem to evaluate the integral ∫C5sin(y)-3y,2+5xcos(y)·dr→, where C is the line…

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

A:

Q: Set up an integral to compute the arc length L of the curve C: r(t) = (, 4t, v2t) %3D from the point…

A: I am going to setup an integral to compute the arc length of the given curve using the required…

Q: 4. Let F = (3y cos(3z) + 4z)i+ (3+ sin(3z))j. Use the Fundamental Theorem of Line Integrals to find…

A:

Q: Find all the values of t where the tangent line to the parametric curve is horizontal or vertical.…

A: x=32t-cos tdxdt=32+sin tvertical tangent line:dxdt=032+sin t=0sin t =-32t=2nπ+4π3  and t=2nπ+5π3,…

Q: ..... The area of the surface generated by revolving the curve x = 1 cos (2t), y = 7+ -sin (2t) on…

A: Follow 2nd and 3rd step.

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. A particle is moving in space with position function R(t) = (sin(3t), cos(3t), 3 – 4t). 2.1. Find…

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Q: Which one of the following is the parametric equa- tions of the line that is normal to the surface…

A:

Q: 1. Find parametrizations of the curve of intersection of x² + 4x + z² = 0 and x- – 2y + z = 3.

A: Given curve is x2 + 4x + z2 = 0 .  This can be written as (x+2)2 + z2  = 4  So the curve is a circle…

Q: that the in tegral is inde pen dent the path, and v6e S how the Fundamental to of of Line Integrals…

A:

Q: The arc length of the curve r= cos 30 with 0<0 ST is I V-9(sin 30)² + (cos 30)² d0 0. I V(sin 30)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.