Determine and state the work done by the vector field over the given path in each situation. F (x,y)= (x² -10xy, 2– 5x) moving from (0,0) to (4,4) to (6, 2).

Q: If a vector field F (x+ 2y + az)f + (bx – 3y – 2)]+ (4x +cy+ 2z) k is irotational then the value of…

A:

Q: F is a path-independent vector field, with F. dr = 5.9 and )F. dr = 2.9 and F. dr = 3.4, find (1,0)…

A: Answer

Q: - draw the vector fields . F(xy) = 2xi – yİk

A:

Q: If the vector field v = [mz+x, -y, x]is irrotational, the mis equal to O A. 2 O B. 0 O C. 1 O D.…

A: Curl of v is zero.

Q: (1) Is the vector field F(x, y) = (2æ*y^ + æ) î + (2a*y³ + y) Î conservative? Support your answer…

A: Note: As per our guidelines we are supposed to answer only one question. Kindly repost other…

Q: 5. Find a potential function corresponding to the vector field F(r,y, 2) = (2r, 3y, 4:).

A:

Q: 1) Find the work done by the vector field F(x, y, z) = (x², 4xy³, xy²) on a particle that %3D…

A: Given force is F→x,y,z=x2,4xy3,xy2 Let r→=rt→ be the boundary of the rectangle with vertices  0,0,0,…

Q: g (x, y) = 5x² + 3xy + y² find the conservative vector field.

A: We know that  a conservative vector field is also a gradient vector. So here we will use this…

Q: Match the vector field with its graph. F(x, y) = yi -6+ 世 -5+

A:

Q: Compute and sketch the vector assigned to the points P = (0, 1, 1) and Q = (2, 1, 0) by the vector…

A: Solution:Given       P = (0, 1, 1) and Q = (2, 1, 0)F=<xy,z2,x>To sketch the graph of F

Q: 41.) Use given grid to sketch the vector field F = (y, -x). Include vectors at all integers points…

A: Given vector field is F=12y,-x

Q: latch the vector field with its graph. F(x, y) = yi -6 5 5 -5+

A: Given: F(x,y)=yi Find: graph of vector field

Q: 7. Determine if the given vector field F is conservative on some domain and, if it is, find the…

A:

Q: → If I is a path-independent vector field, with 00 F. dr = 4.1 and ((11) F · ďr = 3.3 and (0) F · dr…

A:

Q: 11) If f is a potential function for the vector field F = (-2y + 2xyz, -2x + x²z, x² y + 8z), then…

A: To find- If f is a potential function for the vector field F = -2y + 2xyz, -2x + x2z, x2y + 8z, then…

Q: Evaluate , (V. A) dV for the vector field A = x²yi +xz³j- y²z²kwithin the volume of a cube defined…

A:

Q: If a vector field A(x,y,z)=xz°ex - 2x yzey+2yzez , calculate the curl 7xA at point M(1,-1,-1).

A: A vector-valued function is a rule used to fix a vector to a particular point in space. The general…

Q: Given the 2-D vector field : G(x, y) = (2y)î + (x)j and the triangular curve T : line segments from…

A: Given: G→x,y=2yi^+xj^ Triangular curve T, line segments from point A0,0 to point B4,0 to point D4,2…

Q: A vector field is given as G = (1/(√x² + xy))(e¯²ax +xzay), find the unit vector which is normal to…

A: The given vector fields are: G=1x2+xye-zax+xzayP=3x2ax-4yay We have to find the unit vector which is…

Q: Find the divergence in (2,1,-1) for the vector field FCX₁Y₁² ) = x³y² 2 ² + x ² z 3 + x^²yk

A:

Q: 3. Find the divergence of the vector field. F = ye2* i+2 ye³j 2.x²

A:

Q: Find the streamlines of the vector field and the streamline through the given point. F(x, y, z) =i+…

A:

Q: 14. Find the curl of the vector field F = x²i – xevj+ 2xyzk

A:

Q: Given the vector field F(7, y, 2) F (x, y, z) = compute the curl of F and find the values of a and…

A: curl(F)

Q: Determine and state the work done by the vector field over the given path in situation. F (x,y) = (x…

A:

Q: 3. Let F be the vector field (ze²z + yz², xz², xe*z + 2xyz)

A:

Q: 5. Show that F =(y cos(x),sin(x)+ y) is a conservative vector field and use this to calculate the…

A: Given: F → = y cos(x), sin(x) + y. A vector field F→ = P, Q is said to be conservative if partial…

Q: Q5) If the vector field T = (axy + Bz³)a, + (3x² – yz)a, + (3xz² – y)a, is irrotational, determine…

A:

Q: Match the vector field with its graph. F(x, y) = yi -5+ -5+

A:

Q: Determine whether or not F(x,y) = (xy + y²,x² + 2xy) is a conservative vector field.

A: Given vector field is  Fx,y=xy+y2,x2+2xy Fx,y=F1,F2 A vector field is said to be conservative if…

Q: d) Determine whether the vector field is conservative. If it is, find a potential function for the…

A: Let a vector field is given by,                                  The vector field is conservative…

Q: 1 Find the divergence of the vector field F -(xi + yj+ zk) Vx? +y? +Z 2 x² + y² + z? 1 x² + y? + z?…

A: If the vector field is given by F=F1i+F2j+F3k then the divergence of F is ∇·F=∂F1∂x+∂F2∂y+∂F3∂z.

Q: 4. Determine whether or not the vector field F(x, Y, z) = (-xye®, -xe + e° + 2y, 2z) is…

A:

Q: Q4. A vector field F is defined as F = psin pa,+ 2p² zas + zcos oa,. The V × F at point P(1, 7/2, 2)…

A:

Q: Find the divergence of the following vector field: 00 O 2z O F = (y, -x + z², 3x)

A: We can solve this as follows:

Q: Show that if a vector field has the form F = ⟨ ƒ(x), g(y), h(z)⟩, then∇ x F = 0.

A:

Q: Match the vector field with its graph. F(x, y) = xi + 3yj 山供 得 三 -5+ 5 X 三 -5+ ||| 三 5 BE 州 X x

A:

Q: If f is a power function for the vector field F = (2xz, −3z, x2 −3y+12z2), then the value of f(−3,…

A: We have F=2xz, -3z, x2-3y+12z2 Now, F=∇f=∂f∂x, ∂f∂y, ∂f∂z=2xz, -3z, x2-3y+12z2 So, ∂f∂x=2xz…

Q: Select the correct vector diagram for the given vector field: E = xr - yy

A: We are given the vector field : E=x^x-y^y Plotting the vector field we have: Now, if we observe the…

Q: of the vector field Parye" + around the boun 4. and y = 4.

A:

Q: Compute the curl of the vector field F = -2z i – 3y j – 5x k.

A: In geometry and physics and its engineering applications, we use two kinds of quantities, scalars,…

Q: Determine and state the work done by the vector field over the given path in situation. a. i. F (x,…

A:

Q: If a vector field F = (x+2y+ az)î + (bx-3y-z)i+ (4x + cy +2z) k is irrotational then the value of a,…

A:

Q: Q5: a) Show that the vector field F = (x² - y² + x)i - (2xy + y)j, is irrational.

A: Given vector field is F→=x2-y2+xi^-2xy+yj^. To show: Vector field F→=x2-y2+xi^-2xy+yj^ is…

Q: Match the planar vector field with the corresponding plot. (a) (4, x) y -6-

A:

Q: Verify if the following vector fields are conservative: 2y 21 a) A1 = (1+æ²+y² > I+x²+y² b) Ã, =…

A:

Q: If a vector field A(x,y,z)=xz3ex - 2x2yzey+2yz4ez , calculate the divergence ∇·A at point…

A: A vector field is given as  and

Q: 4. Given the vector field Č = p² sin o a, + (z² cos – 1)as + (p² – z)a, Determine the x-component of…

A: Given the vector field

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: