Find the work done by the force F- Tyi + (y - ±)j over the straight line from (0,0) to (3,6).

Q: A force F = -3i +j+6k acts through the point (5,-2,-1). Find its moment about the point (3,0,2).

A: Moment of force about the point is given by cross product of position vector r about a point and…

Q: 5. Find the directional derivative of f(x, y) = xy – y at the point (1, 2) along the unit vector…

A: The given function is: f(x,y)=xy-y2 Given point is: P = (1, 2) Direction vector is: v=3, 4

Q: Consider the function Question 1 x f(x, y) = -y ln x + x lny + y a) Find the directional derivative…

A: Hints: The given function is f(x,y)=-ylnx+xlny+xy The direction vector of a is a⇀a. Therefore the…

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: (2x + 3)y' = y + √2x + 3, when x = -1, y = 0.

A:

Q: Find the diectional derivative of the function f (xiy,z)= sin (xy+2) at the point of P(1, al2 x4) a…

A:

Q: f(x,y,z)=5sin(xyz), (a,b,c)=(1,1,π/4), and v⃗ =(3,-√2,3). Calculate the directional derivative…

A: Here we will evaluate each expression.

Q: Find the work done when a particle moves along the x-axis from x= 1 to x= 3 under the action of a…

A:

Q: B6. Express + ar

A: Given

Q: Find the work done by the force F = xyi + (y − x)j over the straight line from ( − 1,0) to (0, − 1).

A: In the given problem, force is given and the displacement straight line is given. We have to find…

Q: The force exerted by an electric charge at the origin on a charged particle at the point (x,y,z)…

A:

Q: Utilize Green's Theorem to figure out the work done by the force: F(x, y) = x(x + 3y)i + 3xy²j in…

A:

Q: Compute the work done by the force F = (sin(x + y), xy, x²z) in moving an object along the…

A: Given information:Force vector in Newtons.Initial point .Intermediate point .Final point .To find:…

Q: Find r'(t), r(t,), and r'(to) for the given value of to. r(t) = (1 + t)i + t°j, to = 1 r'(t) = r(t,)…

A: Here is the solution of the given problem. Space Curve:

Q: Find the directional derivative of f(x, y, z) = x²y²z + 4xz at (-1,2,1) in the direction 2i + j +…

A: We have to find the directional derivative of f(x, y, z)=x2y2z+4xz at (-1, 2, 1) in the direction…

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: