Let C be the curve with parametrization r(t) : < sin(rt), (t – 2)² + 1 > ,0 < t < 1. Let F be the vector field < - 6y + 2, 6x – 2y > . Find the value of F · dr.

Q: Let CC R² be the circle with radius 8 centered at the origin. Let F: R2 → R² be the vector field…

A: Use the relation, T^=drdtdrdt     Where rt=R cost i^+R sint j^ And N=T^×k^

Q: Consider the vector field F(x, y) = (3y² – 2xy, 6ry – x²). (a) Show that F is conservative. (b) Find…

A: According to question, Given vector field, F→x, y = 3y2 -2xy, 6xy-x2

Q: xi + yj x2 + y2 F(x, y) = k-

A: Topic- vector integration

Q: Find the divergence of the vector field F(x,y,z) = (ye*,2x + 3y, ycosx).

A: Consider a vector field is defined as, Fx,y,z=f,g,h The divergence of the vector field is defined…

Q: Find the divergence of the vector field.

A: To Find the divergence of the vector field.

Q: Find (a) the divergence of the vector field and (b) the curl of the vector field.

A: Given :   To find : a) The divergence of the vector field. b) The curl of the vector field.

Q: Consider the vector field F = (5x + 6y, 6x + 4y). F - dr along the curve C given by r (t) = (t², t*)…

A: ∫CF→·dr→=∫abF(r(t)) r'(t) dr

Q: Consider the vector field F = (2y +1,y, –2²) along with the paraboloid surface z = r² + y. Find the…

A: Circulation is the line integral of the vector field along the curve

Q: Find the curl of the vector field F. F(x, y, z) = (2y - z)i + e²j + xyzk

A:

Q: Find the curl of the vector field F(x, y, z)= e sin yi - e cos yj.

A:

Q: Find the divergence of the vector field F. F(x, y, z) = In(9x2+ 6y² )i + 54xyj + In (6y? + 5z?)«…

A: To find the divergence of given vector field

Q: For the vector field F(x, y, z) = (15x2y + 2xz)i + (5x3 + 35y4)j + x2k, show that F is conservative…

A:

Q: Consider the vector field F(x, y, z) = (2ry, x², 1). (c) Calculate the line integral / F - dĩ where…

A:

Q: Find the divergence of the vector field F. F(x, y) = 4xe*i - 7x?y²j

A: Find the divergence of the vector field F.

Q: Given the following vector field and orientated curve C, evaluate F•Tds. (x,y) F = on the curve r(t)…

A:

Q: Find the work done by the vector field F = (– 4z, – 3y, – 2x) in moving an object along C in the…

A:

Q: 2.) Calculate the curve integral over the vector field V(17, y) = ( * -2²y xy? for the positive…

A: Disclaimer: Since you have asked for a handwritten solution and currently we are not providing any…

Q: Consider the vector-field F(x, y) = y²i-xj, and let C be the arc of the parabola y = k – x² from the…

A:

Q: Determine whether F(x, y) = xi +9yj is a conservative vector field. If so, find a potential function…

A:

Q: Verify if the vector field F = (x° cos(4y), -x sin(4y)) is conservative and evaluate the line…

A:

Q: Find the divergence of the vector field F→(x,y,z)=7sinxi→−cos(3y)j→+z^9k→.   Enter the exact…

A:

Q: Find the divergence of the vector field F = < yx², xzª, zy³ div F =

A: f = y x2  ∂f/ ∂x = 2xy   g = x z4  ∂g/ ∂y = 0   h = z y3  ∂h/ ∂z = y3

Q: Show that the vector field F(x, y, z) = (le* cos y + yz)i+(xz - e* sin y)j+(xy + 4z) k is a…

A: Given: F(x,y,z)= (excosy+yz)i+(xz-exsiny)j+(xy+4z)k To show that F is conservative and if it is,…

Q: Find the divergence and the curl of the vector field .F(x, y, z) = 1 (xi + yj + zk).

A: F(x,y,z) =xi +yj+zkx2+y2+z2 it can be written as F(x,y,z) =xi +yj+zkr   where r =x2+y2+z2 divergence…

Q: Find the curl of the vector field F = (4x sin(y), 7y cos(x)). curl F - Dretieur

A:

Q: Show that the vector field given below is conservative and find its potential function. F(x, y) =…

A:

Q: Match the vector field with its graph. F(x, y) %3D хі + Зуј -67

A:

Q: find the work done by the given vector field F along the specified curve C.   F(x, y) = yi − x j,…

A: Given, F(x,y)=yi^-xj^ where curve, x2+y2=4 W=∫F.dr W=∫ydx-xdy from curve, dx=-dy W=∫xdx+ydy limit of…

Q: Given a scalar function V(x, y) =-(x² +y² ), the directional derivative of V in the direction of the…

A:

Q: Consider the vector field F = (r²y³, x°y?) Is this vector field conservative? Conservative If so:…

A: Given the vector field &lt;x2y3,x3y2&gt;. We need to: i) Check whether the vector field is…

Q: Let F be the vector field (sin x,ey²) and C be the ellipse x? + 5y² = 1, oriented counterclockwise.…

A: Green's theorem

Q: Consider the vector field F = 2xy i + (x2 + 2y) j. Verify the Green's Theorem in the closed region…

A:

Q: Ex: Find the work done by the vector field F = (y-x²)i + (z - y²)j + (x-z²2)k Over the curve r(t) =…

A: Given: F=y-x2i+z-y2j+x-z2krt=ti+t2j+t3k

Q: Let F be a vector field and Ca curve with parametrization 7 (t) where a <t<b. F. dr = | (cos(t³),…

A:

Q: Find the curl of the vector field F < ya?, xz°, zyš curl F = k || + +

A:

Q: Find the divergence of the vector field F. F(x, y) 4xe*i 6x²y?j

A:

Q: Find the line integral of the vector field F= over the curve C which is the quarter of the unit…

A: Given: vector field is F→=y-x, x. To be find: line integral of the vector field over the curve C…

Q: Use the divergence of the vector field to compute F(x, y) = (tan-1 (2²)) i + (xy-2 +82²) j 4 fo…

A:

Q: Find (a) the divergence and (b) the curl of the vector field F(x, y, z) = cos(z)i + 5ysin(5z)j+…

A: Given vector field is  F(x,y,z)=(cosz)i+5ysin(5z)j+2x2zk

Q: Show that the vector field F = 2.xyi + (x – 2 y)j is conservative and evaluate the line integral %3D…

A:

Q: Let be the vector field defined on the plane by F(x, y) = (x¹+3y, 6x - siny).

A:

Q: Find the divergence of the vector field F. F(x,y,2) %3D 6хe\i + 4ye'j

A:

Q: Consider the vector field F and the curve C below. F(x, y) = (6 + 8xy²)i + 8x?yj, C is the arc of…

A: It is given that F→x,y=6+8xy2i+8x2yj. In F→x,y=6+8xy2i+8x2yj, P=6+8xy2,Q=8x2y. Obtain ∂Q∂x, ∂P∂y:…

Q: Let D be the upper half-plane (y > 0) and consider the vector field F : D+R² given by F(x, y) = (…

A:

Q: Consider the vector field F and the curve C below. F(x, y) = x2y³i + x³y?j, C: r(t) = (t2 – 2t, t2 +…

A:

Q: Given the vector field F(x,y,z)= , find the divergence and the curl of F.

A:

Q: Match the vector field with its graph. F(x, y) = xi + 3yj KE 、 ||十

A:

Q: Consider the vector Field 7 and the curve C below. F(x,y) = (3 +4 ху?) 1 + 4x²yј, C is the arc of…

A:

Q: Given vector field F(x, y) = (4x³y? – 2xy³,2x*y – 3x?y² +6y³) and curve C defined by C: 7(t) = (t+…

A: Here we will find out the required value.

Q: Consider the vector field F and the curve C below. F(x, y) = (4 + 2xy²)i + 2x²yj, C is the arc of…

A: