5. Given the vector field F(x, y, z) = (e", (xe" + e²), ye²) find the following a. Show F is conservative b. Find a potential function f (i.e. F = Vf) c. Find the line integral S. F- dř where C is the line segment from (0,2,0) to (4,0,3) %3D

Q: Find work done by the force field f = (XY, 2, 42) when a Particle moves along the straight line from…

A: The force field is given as,find work done by the force field F when a particle moves along the…

Q: 5. Consider the force field F = (xy)i + (x −y)j and C, the straight line from the point (-1,2) to…

A:

Q: Find the wark done by the Foree Field FoK x>ye^> along the parabola X=y+1 from (bo) to (2,1).

A:

Q: Find the electric flux PE across the part of the cylinder x² + y² = 9,0 ≤ z ≤ 2, in the direction of…

A: Given information:The electric field is given as .The surface is a part of the cylinder defined by…

Q: Let r(t) = (e", e", te") be a vector function. Find; 21 -2t (a) r'(1) (b) r"(1) %3D

A:

Q: Sketch the vector field F(x, y)=2i-j and calculate the line integral of along the line segment from…

A:

Q: Find r'(t), r(t), and r'(to) for the given value of to- r(t) = (e², e²t), to = 0 r'(t) = r(to) =…

A: rt=et,e2t,t0=0

Q: Let A be the vector potential and B the magnetic field of the infinite solenoid of radius R. Then

A:

Q: 1. Find the line integral in a vector field a curve consisting of 2 parts. First, the straight line…

A: Given F→=y,x+2y We have to find the ∫CF→⋅dr along the given path.

Q: Determine the line integral if the vector field is F(x, y) = x²yî – 2xyf On a rectilinear path that…

A:

Q: Suppose F(x, y) = x*i + yj and C is the line segment segment from point P = (-5, 5) to Q = (-8, 9).…

A:

Q: Find the work done by F over the curve in the direction of increasing t. F = 4zi + 4xj + 4yk, r(t) =…

A: Note that : Since you have posted multiple questions, we will provide the solution only to the first…

Q: Show that the force F = (y^²2² - 6xz² ) ₁² + ²xyz²³ j^² + (3 xy² z ²-6x²²Z) k IS Conservative, hence…

A:

Q: 5. Compute the directional derivative and gradient of f(x, y, z) = 4x – y²e3xz at (2, – 1, 0) in the…

A: Gradient of vector function For a given vector f(x, y, z) the gradient vector denoted by ∇(f) is […

Q: 6. Let f(x, y) = ln(2x - y²). (a) (b) (c) Determine the gradient of f at the point (2, -1).…

A:

Q: The acceleration of a particle at any time >=0 is given by: dr a = == (25cos2t)i + (16sen2t)j +…

A:

Q: Find the work F dr done by the force field F = (y, x, z) on a particle that moves along C the line…

A:

Q: Let A be the vector potential and B the magnetic field of the infinite solenoid of radius R. Then 0…

A:

Q: Find the outward flux of F = (4x +25y*,0,) 10z through the surface 25 4 9

A: Solution:-

Q: Given the vector fieldF(z,y)=(y³+422-2)i+(3zy2+2y+2)j. Evaluate the line integral SF-dr from (1,0)…

A:

Q: A conservative vector field is given as F = 6xe'i + (3x²e" –10yz')j-15y²z°k. Find the potential…

A:

Q: 5. For f(x, y) 2x +y v 3i-4j find the rate of change of fat (1,2,6) in the direction of the vector =…

A: rate of change of the function at the given point along the given direction

Q: 1. Let F =i+y^2j +z^4k. Find the general flow lines for this vector field. Present the flow line…

A: Consider the provided question,

Q: Sex. (x+y)ds where C is the straight-line segment x = 3t, y = (3-3t), z = 0 from C (0,3,0) to…

A:

Q: The work done by the vector field F(x, y) = (-y², ²) along the arc of the parabola y = 2x² - 4x + 2…

A: According to question,Given Vector field,Fx, = -y2,x2Equation of pparabola,y = 2x2-4x+2

Q: Find the potential function f for the field F = 6x i + 6y j + 7z k. A general expression for the…

A: Note: As you have posted multiple question but I will answer first question according to the…

Q: 3. (a) Given the vector valued function r(t) = 4 cos zti + 4 sin atj + tk. Find the position vector,…

A: Since you have asked multiple question, we will solve the first question for you. If you want any…

Q: Compute the integral of ſA° · dľ°P2P1 from point P1 (2,1, -1) to P2 (8,2, -1) of the vector function…

A: We have to find the integral from point p1 TP p2.

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: