(5) Evaluate F- dr using Stokes' Theorem, where F(1, y, 2) = -y?i + xj +2²k and C is the curve of intersection of the plane y+z = 2 and the cylinder r² + y² = 1 (Orient C to be counterclockwise when viewed from above) as shown below; -y+z=2 D OL

Q: a) f : R? → R, f (x, y) = Vy sin x 5) f : R2 → R. f (r. u) = GGrul + In (a2 + 1? – 1). /4 – 72 -

A: We will find the domain

Q: What is the parametric equation for the line of intersection of the planes x + y - z = 2 and 3x - 4y…

A:

Q: ind the equation of the plane tagent to the curves f (t) = [3t+4,4 tº, sin (πt) + 4] and g(s) = [8°…

A:

Q: E.X; Let T = {E₁ = (-∞,a): a = R}U{R,Ø} be a topology on R and let A = [2,3), B=(3,4), C = (1,5], D…

A: It is given that,  T = Ea=-∞, a : a∈ℝ∪ℝ, ∅ is a topology on ℝ. We have to find the closure of the…

Q: Let L be the line with parametric equations x = 2+t, y = 5-2t, z = 4 + 3t. Let S be the plane that…

A: Given L is the line with parametric equations x = 2 + t , y = 5-2t , z = 4 + 3t . Let S be the plane…

Q: 1. Let R and b be positive constants. The vector function r(t) = (R cost, R sint, bt) traces out a…

A:

Q: 4. Consider the vector function r(z, y) = (x, y, x² + 2y*). (a) Re-write this vector function as…

A: For the given vector function, PART A The surface function is given by, f(x,y)=xi^+yj^+(x2+2y2)k^…

Q: A communications satellite orbiting the earth has initial posi- tion r = (29,000, 20,000, 0) (in…

A: GIVEN  we need to find the equation of plane containing the satellite 's orbit

Q: (b) Calculate the distance from the point P(3,–1, 2) to the plane with equation 2x – y +z = 4.

A:

Q: 1. Obtain the directional derivative of: a.. f(x,y) = x²-4x³y² at the point (1,-2) in the direction…

A: Step 1:

Q: 7. For the vector-valued function √t-1 ln(2t - 1) t² - 1 r(t) = (VI- t-1 2 COS πt rt), (t (t = 1):…

A: Let us solve above problem

Q: Find the points on the surface 1x² + 1y² + 3x² = 1 at which the tangent plane is parallel to the…

A: We know that two planes are parallel if their normal vector are parallel. That is their, gradient…

Q: Where does the plane tangent to z = At z = 5exy at (7, 7, 5) meet the z-axis?

A: We find equation of tangent plane to z=5e^(x-y) at (7,7,5)

Q: B-For the surface z² = x² + y² find the directional derivative at the point (√2, √2,-2) in the…

A:

Q: Let P be the point (2, -1, 2) and Ø,: x² + y² – z² = 10. (i) Find the unit normal vector to the…

A: (c) The given surface is ∅1: x2+y2-z2=10 and the point P: 2,-1,2. (i) The unit normal vector to the…

Q: 16xydx- 24ydy = -8x²dy %3D

A: The given differential equation can be written as:16xy−24yy' =−8x2y' An ODE Mx, y+Nx, yy'=0 is in…

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: