c) Verify Stokes's Theorem for F = (x²+y²)i-2xyj takes around the rectangle bounded by the lines x =2, x=-2, y=0 and y=4

Q: Find an equation of the normal line to the parametric surface given by R(u, v) = ((2+ cos v) cos TU,…

A:

Q: 03. The equation of the tangent plane to the surface f(x,y) at the point (1,1) is 2x + 2y = 9, find:…

A: We have given equation of tangent plane 2x+2y=9which is incorrect because left side is even number…

Q: Use Stokes's Theorem to evaluate Jo F. dr. In this case, C is oriented counterclockwise as viewed…

A:

Q: Verify Stokes's theorem for F =, where C is the curve C: x² + y² = 4, z = 2

A:

Q: Evaluate the unoriented line integral (22 + y° + z²) ds (2x² along the parametric curve a : [0, 1]…

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: r(u, v)=(w+v)i(u-v)j+v²k at the point P(2, 0, 1)

A:

Q: Verify the validity of the Stokes's theorem for the vector A = e-bz sin ø a, + pe-bz , over the open…

A:

Q: Q2) Evaluate each side of Stokes theorem for a portion of sphere :r=2m, O=t/3, 0<¢<a H= 2r cosO sino…

A: This question is about application of Stokes theorem

Q: A surface z = f(x, y) is defined implicitly by the equation x +y+ z – e² = 3 – e. - Determine the…

A:

Q: 1. What is the differential for the total surface area of a frustum of a cone with l as the slant…

A: Given slant height l. Let r and R be the radii of the frustum. Let h be the height of the frustum.…

Q: Evaluate the line integral / V(ry + xz + yz) · dĩ over the curve F(t) = (t, 2t, 3t) for 0 <t < 4…

A:

Q: Evaluate the line integral for the given curve C. S-ds, C: r(t)= (t',t*),1<t<2 cy ||

A:

Q: Use Stokes's Theorem to evaluate F(x, y, z) = x²i + z²j – xyzk S: z = √√√4x² - y² F. dr. In this…

A: Consider the given function, Fx,y,z=x2i+z2j-xyzk and S:z=4-x2-y2 Let us consider that ϕ:x2+y2+z2=4.…

Q: 4) Find the equation of the tangent plane to the surface r(u, v) = (u²-v²)i − (3uv)j + (4√u+v)k when…

A: Note:- As per our guidelines, we can answer the first part of this problem as exactly one is not…

Q: Give the integral representing the length of the parametric curve x=t^3, y=t^4, 0≤t≤1.

A: Given: A parametric curve:   Now the length of the parametric curve is given by the formula:

Q: b) Find equations for the tangent plane and normal line at the point po on the given surface.…

A:

Q: Find the equation of the tangent plane to the surface z = ln (2x + y) at the point (-1,3,0) .…

A: To find the equation of the tangent plane to the surface z=ln2x+y at the point -1, 3, 0.

Q: Use Stokes's Theorem to evaluate JeF - dr- . In each case, C is oriented counterclockwise as viewed…

A:

Q: Suppose F(x, y) = -y sin(x)i + cos(x)j. x? from the origin to the (a) Find a vector parametric…

A:

Q: Find equations for (a) the tangent plane and (b) normal line at the point P on the given surface 1)…

A:

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: Suppose F(x, y) = -y sin(x)i + cos(x)j. (a) Find a vector parametric equation for the parabola y =…

A:

Q: STOKES' THEOREM o may be parameterized by r(r, y) = (1, y, f(x, y)) = curl F

A: We shall answer first three subpart only as per answering guidelines. For others, kindly post again…

Q: Use laplace tansform y"+ + 4y" +qy'+1Oy= - 24+e y1o) =0, yl0) = -4 , y"O) -1o %3D

A: We use Laplace Transform to find function in t

Q: Consider the function f(x, y) = 2xy + x2 + 5 Find the equation of the tangent plane to the surface…

A: given: f(x,y)=2xy+x2+5 find the equation of tangent plane to the surface z=f(x,y) at (-1,1,4)

Q: Verify Stokes's theorem over a quarter circular disk with a radius, r = 4 in the 1* quadrant for any…

A:

Q: Use Stokes's Theorem to evaluate F(x y₁ z) = x² +2 - xyzk F. dr. In this case, C is oriented…

A: The solution is given below

Q: Use Stokes's Theorem to evaluate F. dr. In this case, Cis oriented counterclockwise as viewed from…

A: We are given a line integral and we have to use Stoke's Theorem to evaluate that integral.

Q: What is the differential for the total surface area of a frustum of a cone with i as the slant…

A: The formula for total surface area of the frustum of the cone with the given slant height l is Total…

Q: 4. Let g(x, y) = e"/v, Find an equation for the tangent plane at (x, y) = (2, 1).

A:

Q: Use Stokes's Theorem to evaluate F(x, y, z) = z²i+yj + zk S: z = √√16x2 - y2 6 F. dr. In this case,…

A:

Q: Use a parameterization R(u, v) of f (x,y) = xe*y to find the tangent plane to f(x, y) at the point…

A: Tangent plane at (x1, y1) to f(x, y) is given by below:

Q: Use Stokes's Theorem to evaluate F(x, y, z) = z²i+yj + zk S: z = √4x² - y2 F. dr. In this case, C is…

A:

Q: Let f (x, y, z) = ln (1+x³ + y* + z®) Find an equation for the tangent plane to the surface f (x, y,…

A: We have to find the tangent and normal line equation .

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

A:

Q: Find the area under the curve for the parametric function defined by the equations x(t) = 5t, y(t) =…

A:

Q: Clearly demonstrate how you use Stokes' Theorem to evaluate f F 1- y³, x³ + e²,0) and C is the…

A:

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: 5. Show that the equation of the tangent plane to f(r, y, z) = c at (ro, Yo: 20) is given by the…

A: This is the question from vectors.

Q: A force F acts on a particle that is moving in the plane along the semi-circle parameterized by 7(t)…

A: Work done W=∫CF.ds=∫abF(r(t)) . r'(t)dt   Where curve C is given by parametric equation r(t)

Q: Suppose F(2, y) = -y sin(æ)i + cos(2)j. (a) Find a vector parametric equation for the parabola y =…

A: Given F→(x,y) = -ysin(x)i→+cos(x)j→ a) Let us assume x=t then y=x2=t2 by differentiating r→(t) with…

Q: State the Stokes' Theorem. Ise the Stokes' Theorem as stated in (a) to evaluate the lin |F- dř here:…

A: We have to write the stokes theorem and using it We have to find the given line integral according…

Q: Verify Stokes' theorem for F = z²i – y²j and the curve C is the square in the xz-plane from (0,0, 0)…

A:

Q: Use Green's Theorem to evaluate the line integral. Orient the curve counterclockwise. e5x + 2y dx +…

A:

Q: Let F = (5x4 + 16z, 2x³ + 3y? + 16z, x + y+ ev*+1) and C be the curve defined by 7 (t) = (2 cos(t),…

A:

Q: Let F = ( ) 5x* + 16z, 2x³ + 3y² + 16z, x + y + eVz³+1 and C be the curve defined by 7(t) = (2…

A: Given that,  F=5x4+16z, 2x3+3y2+16z, x+y+ez3+1 And rt=2cost, 2sint, 16cos2t sin2t, o≤t≤π To find…

Q: 4. Use Stokes Theorem to find the circulation of the vector tield F around any smoo th, Simple…

A: Stokes theorem ∫CF.dr→=∫∫Scurl F.N→ ds Curl of the vector point function Ff1, f2 f3…

Q: Assuming that the equations define y as a differentiablefunction of x, use Theorem 8 to find the…

A: the equation we have  (x3-y4)6+ln(x2+y)=1

Q: Find the coordinates of the centroid of the curve x = cos t + t sin t, y = sin t - t cos t, 0…

A: Given,

Arc Length

Arc length can be thought of as the distance you would travel if you walked along the path of a curve. Arc length is used in a wide range of real applications. We might be interested in knowing how far a rocket travels if it is launched along a parabolic path. Alternatively, if a curve on a map represents a road, we might want to know how far we need to drive to get to our destination. The distance between two points along a curve is known as arc length. 

Line Integral

A line integral is one of the important topics that are discussed in the calculus syllabus. When we have a function that we want to integrate, and we evaluate the function alongside a curve, we define it as a line integral. Evaluation of a function along a curve is very important in mathematics. Usually, by a line integral, we compute the area of the function along the curve. This integral is also known as curvilinear, curve, or path integral in short. If line integrals are to be calculated in the complex plane, then the term contour integral can be used as well.     

Triple Integral

Examples: