Given the vector field H = pz cos o a, + sin ap +p²az and a point Q 1,,0), solve the following and express in cylindrical components. The vector component of H that is normal to the surface p = 1 at the point Q The vector component of H that is normal to the surface z = 0 at the point Q a.

Q: Find the positive t when the vector r(t) = (4t, 4ť, 5t² – 6) is perpendicular to r' (t). -

A:

Q: Ans. (5) Given position vector r(t) = (t₁ t - sint, 1- cast) ⇒ r (t) = ti + (t-sint) 3 + (1 - cost)…

A: Ans. (5) Given position vector

Q: 5. Find the parametric equations of the line tangent to the curve C defined by the vector equation…

A: To find the parametric equation of the given tangent to the curve. The vector equation of the curve…

Q: Give a vector parametric equation for the line that passes through the point (3, 5, –5), parallel to…

A:

Q: 2) Let two vector functions be defined by r₁(t) = (2t, t², t²) r₂(t) = (sint, sin(3t),5t) a. Find…

A: a. Consider the given vector functions, r1(t)=2t,t2,t4r2(t)=sint,sin3t,5t if two vector function…

Q: Find T(t) ,N(t),B(t), and K(t) for the vector position r(t) = (e2t cos t) i + (e2t sin t)j + 2k when…

A: Given rt=e2tcosti+e2tsintj+2k

Q: 12. Given the vector equation [x. y] = [2.-3]+ (4.5] find a) the parametric equations b) the…

A: Given the vector equation [x, y]= [2, -3] + t[4, 5] find (a) Parametric equations (b) Symmetric…

Q: a) Find the scalar parametric equation and vector equation (r = c + ta) of the straight line z+5 x –…

A: We have to find

Q: Find the position vector for a particle with acceleration, initial velocity, and initial position…

A:

Q: Consider r(t)=⟨sin(t),cos(t),2t⟩ the position vector of a particle at time t∈[0,3], where the…

A:

Q: A spring gun located at the origin O = (0,0,0) fires a ball at time t = 0 at an angle of 30°. The…

A: First write the general position vector. Now determine the points through which it passes and the…

Q: 1.Solve the following polynomial equations.  a)  2X3+1=x2+2X

A:

Q: (a) Find a vector function that describes the curve of the intersection of the cylinder z²+²=4 and…

A: We will use trigonometric identity to find parametric equation of curves and combined them to find…

Q: A fire ant, searching for hot sauce in a picnic area, goes through three displacements along level…

A: Since you have posted multiple questions. I am answering the first three sub-parts. Please post the…

Q: An airplane took off from point (3, 0, 0) at an initial velocity, v(0)= 3j with an acceleration…

A: Let r(t) be a position vector of a particle at time t. Then the velocity vector  v(t)=ddtr(t) Also…

Q: Ben can wash a car in 2 hours while his younger brother James can do it in 3 hours. How long will it…

A: In 1 hour Ben can wash 12 part of car. in 1 hour his younger brother can wash 13 part of car.

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: