Use Stokes' theorem to evaluate      CF · dr  where C is oriented counterclockwise as viewed from above. F(x, y, z) = <−yx2, xy2, exy >    C is the circle in the xy-plane of radius 4 centered at the origin

Q: Find the equation of the tangent plane Pr to the surface S: x2 + y? +z² = 9 at (2, –1, 2)

A: The equation of the tangent plane to the surface f(x, y, z)=0 at a, b, c is defined as: fxa, b,…

Q: Estimate the area under the line y= 2x+1 using six rectangles and right endponits over the interval…

A:

Q: find the tangent plane to the surface z = Vy-x at the point (1,2,1)

A: Equation of tangent plane is z-c = df/dx(x-a)+df/dy(y-b) at point (a,b,c)

Q: Find the equation of a tangent plane to the surface of z = eª cos(y) at the point P(0,0, 1).

A: Consider the provided question, Hello. Since your question has multiple parts, we will solve first…

Q: Use Stokes' theorem to evaluate Ĝ - dĩ, vhere Ĝ(1, Y, 2) = (2r, y,0) and D is any closed curve.

A:

Q: Find an equation of the tangent plane to the parametric surface R(u, v) = ((u - sin u) cos v, (1 -…

A:

Q: Let fcxry)= 2X sinly)+3X Cos CY ). Find the tangent Plane at the point 2,0).Use the tangent Plane to…

A: In this question, concept of tangent plane at any point is applied. Tangent Plane Naturally, it…

Q: = Let x: R? → R³ be the function given by x(u, v) = (u cos v ,u sin v , bv) 1. Check whether x is…

A: (1)Given that:x:R2→R3 is the function given by x(u,v)=(ucosv,usinv,bv)In order to check the proper…

Q: Use Green's Theorem to evaluate fa F. dr where F(x, y) = (y cos x, x² + 2y sin x) and C is the…

A:

Q: Which of the following is the tangent plane to z = 2x² + y² at the point (1, 1, 3)? z = 4x + 2y z =…

A:

Q: Find the equation of the tangent plane Pr to the surface S:x2 +y? + z2 = 9 at (2,-1, 2)

A: To find :Equation of tangent plane to the surface x2+y2+z2=9 at the point (2,-1,2).

Q: 7. Let F(x, y,z)=-2yi +2xj. Use Stokes’ Theorem to find F dr where C is a circle (a) Parallel to the…

A: By stoke's theorem

Q: Use Green's Theorem to evaluate Sc F. dr, for F(x.y) = (y cos x – xy sin x, xy+x cos x ), and C is…

A: Given, F→=ycosx-xysinx,xy+xcosx=ycosx-xysinxi^+xy+xcosxj^

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: