Question 31 Find the flux of F through S, F NdS where N is the upward unit normal vector to S. F(x, y, z) = xi+yj+zk, S: z = 36–x² - y², z 20 972 T O 1,944 T O 2,527 TT O 1,361 7 O 583 77

Q: Let t be a fixed real number and let c = cos t, s = sin t, x = (c, cs, cs2, . . . , csn−1, sn)T Show…

A: If x is a unit vector then:

Q: 1.9 Perpendicular unit vector* Find a unit vector perpendicular to A = (î + j- k) and B = j- 3k).…

A:

Q: Let N = (p, q, r) be the unit normal vector at the given point. %3D r(t) = (cos t, sin t, t),…

A: Given Data The unit normal vector is N=<p,q,r> The point is (1,0,0) The vector is r(t)=<cos…

Q: H.W: Let V = {(x, y): x 20} with standard operation on R2. Is Va vector space?

A: A vector space is a set V that satisfies the following 8 conditions. Here two operations "+" and "."…

Q: Which of the following is the directional derivative of f(z, y) = at the point (0,0) in the…

A:

Q: Find a unit normal vector of the surface at the given point P. x² + y² = 25, P: (4, 3,6)

A: The equation is x2+y2=25 and the point is P4, 3, 6. Now, the unit normal vector is n^=∇ϕ∇ϕ. Now, for…

Q: Find the normal vector N(t) to r(t) =〈4,sin 2t, cos 2t〉.

A: Given: rt=4, sin(2t), cos(2t)

Q: %3D Given ü = 7ỉ – 3+k and w = -2i +5j – 6k compute proj „ .

A:

Q: Find the tangential and normal components of the acceleration vector. r(t) = (3t – t³) i + 3t² j O…

A: Take derivative.Find the magnitude.

Q: 33. Define T: R³ → R³ by y + 2z 2x +3y - z x +2y - 2z x - y a. Find all vectors in R' that are…

A:

Q: What is the principal unit normal vector for the curve: r(t) = (3t + 1)i + e'j .at the point P = (1,…

A: Let r(t) be the curve, then the unit tangent to the curve is given by T(t)= r'(t) / |r'(t)|. The…

Q: 2. Find a unit normal vector to the surface x + y +2 = 5 at the point (0, 1, 2). Ans. Û + 2k)

A:

Q: Haw.y If the directionl derivative Valne ( Du f) of the function flx,4) = x-3xy +uy 4t Paint (-2,0)…

A:

Q: Find the projection of the vector v onto the vector w if v = 2i + j and w = 6i + 3j

A: we have given v and w vectors . We have to find the projection of the vector v onto the vector w

Q: Find the unit tangent vector, unit normal vector, and unit binormal vector given R(t) = ti+t²j+t³k…

A:

Q: . Consider the force vector F = x²a,-xza, -y²a,. Calculate the work done in moving an object from…

A:

Q: The Acceleration , the Unit Tangent Vector and the Principal Unit Normal of a moving particle in…

A:

Q: Find a unit vector in the direction in which f(x, y, z) = 3x³z² + 3y³z +24z - 3 increases most…

A:

Q: Let à = ( − 1, 3, — 5) and 7 = ( − 3, 2, 3). - Find a unit vector normal to a and b with negative z…

A:

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: