Let F = Vx, 0, -/z] be a vector field and C be the curve given by the parametrization r(t)= [cos2(t), 0, sin (t)], where 0

Q: Given the vector field E = 4zy² cos 2xa, +2zy sin 2xay + y² sin 2xa₂ for the region [x], [y], and…

A: The general form of a vector-valued function in three dimensions is given by F=F1ax+F2ay+F3az, where…

Q: What is the (numerical) value of the 3-form field (x² + y + z?) dx A dy A dz on the ordered triple…

A:

Q: 3. Let y be the oriented curve in R³ parametrised by t+ (t2, sin t, cos² t) for 0 <t <n. Let F be…

A:

Q: Show that F is a conservative vector field and use this fact to evaluate F. dr along the given curve…

A: Consider the given function. Fx,y=x2i+y2j The compare the form of the function with the general…

Q: 6. Let F = (x³ sin y, a* cos y). COS (a) Is F a conservative vector field? (b) If your answer to…

A:

Q: Find the value of the line integral ∫CF→⋅dr→if F→(x,y)=(y^3+11)i→+(3xy+11)j→is a conservative vector…

A:

Q: Q3 Given the conservative vector field, F =(x +y²)i +(2xy +5)j , evaluate the integral F•ds where C…

A:

Q: Q3 Given the conservative vector field, F = (x +y²)i +(2xy +5)j , evaluate the integral F•ds where C…

A:

Q: 2. Given that F(x, y, z) = (e", xe" + e, ye²) is a conservative vector field. a. Find a potential…

A: Here we find the potential function using the partial derivatives and integral as appropriate. Then…

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

A:

Q: 8. The component of the field F= (1,2,3) normal to the plane z = -3x + 3y -1 is (a) 0. (b) –2//19.…

A: If we are given a plane equation in the form Ax+By+Cz=D, then the normal vector to the plane will…

Q: For Parts (a) (b), you are given a vector field and a curve C. Use the Fundamental Theorem of Line…

A: Solution is given below:

Q: Let a vector field F(х,у) 3sinx 1 + ху ј be given. What is the integral representing the work done…

A: Workdone by force along the path is integration (f.dl) Where f is force and dl is distance vector…

Q: Evaluate the line integral F. dr, with vector field F(x, y, z) = sin xi + cos yj + xz k, and path C…

A:

Q: (a) Give the definition of line integral of a vector field over a path. (b) Compute the line…

A: Use line integral concept

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: The circuitation of the vector field F(x, y)- (3 + xy´, 4x + xy) along the boundary of the plane…

A: The solution are next step

Q: Consider the surface integral of the vector field F = [xy², 3x², x²z] over a surface S, where S…

A:

Q: 9. Show that F is a conservative vector field and use this fact to evaluate f F dr along the given…

A: Given information: Fx,y=y2i+x2j C is the arc of the parabola y=2x2 from (-1,2) to (2,8). Gradient…

Q: 6. Let F = (2xy)i + (x²)j be the force field Use the Fundamental Theorem for Line Integrals to…

A: The given force field is F= 2xy i+x2j A=(-1,2) B= (3,1) The fundamental theorem states that a line…

Q: Find the work done by the field F(x,y, z) = (6z, 12x, y?) along the curve r(t) (cos(t), sin(t),…

A:

Q: Let F = (a*y+ sin(2), --4æ°y² + 3, æyl8 ) be a vector field on IR³. Find a vector potential G = (g1,…

A: There are so many choices of G.

Q: Consider the vector field F = (x² + y², 7xy). Compute the line integrals f. F· dr and f F· dr, where…

A: Given F=x2+y2,7xy, c1t=t,t2 and c2t=t,t for 0≤t≤1. We know that ∫cF·dr=∫abFxt,yt·C'(t)dt. When…

Q: The vector field F=(x+2y)i+(2x+y)j is conservative. Find a scalar potential f and evaluate the line…

A: Given function is, F=x+2yi+2x+yj

Q: 3. Consider the vector field F(x, y) = (2xy, x² – sin y). Evaluate %3D F. ds where C is the curve…

A:

Q: 2. Letr =√x² + y² + z². (a) For n ≥ 1 and r> 0, express V(r-n) in terms of r and the radial vector…

A: Given  solution(a): Radial vector field  solution(b):

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: 4. Evaluate the line integral Vf-r'dt, where the curve C is given by the vector function r(1)…

A:

Q: In(t) i + t312j + t cos(at)k, F. dr of vector field F(x, y, z) = 3x²zi + zj + (x³ + 2yz)k along…

A:

Q: 1.) Given the conservative vector field F(x,y,z) = ( y In z,x In z, xy find the work required to…

A: The conservative vector field Fx,y,z=y ln z, x ln z, xyz. The parametric equation of line segment…

Q: Find all potential functions of the conservative vector field F (x, y, z) = (2xz + y², 2xy,x² + %3D…

A:

Q: Consider the following vector fields: F = (3x² e²y, 2x³e²y + 2 cos y sin y) and G= (y+z, x+y, z —…

A:

Q: Consider the vector field F=(x^2+y^2,10xy). Compute the line integrals ∫c_1 F⋅dr and ∫c_2 F⋅dr,…

A: Given:

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

A:

Q: (a) Show that the vector field F(x, y, z) = (1 + 3x²y)i + x³j is conservative and find the potential…

A:

Q: 6. Determine whether the vector field is conservative and, if so, find a potential function. a. F(x,…

A:

Q: Calculate the work done by the field F when the object moves along the given path from the initial…

A:

Q: 19. (a) Evaluate the line integral F. dr, for F(x, y) = ex−¹ i + xy j and C is given by r(t) = t² i…

A: (a)F(x, y) = ex-1 i + xy j   And C: r(t) = t2i +t3j , 0≤t≤1.⇒drdt = 2t i + 3t2j and F(r(t)) = et2-1…

Q: Consider again the vector field F = (x + y)i + (x² + y² )j. What is the downward flux across the…

A:

Q: Show that the vector field F = 2.xyi +(x² – 2 y)j is conservative and evaluate the line integral…

A: Given the vector field

Q: Consider the conservative vector field 4 8 F(x, y, z) = – 3y²cos (3nz),37y sin (3tz) – x2). 2xz, %3D…

A:

Q: Find the path integral of a particle moving along a straightline connecting O(0,0,0) to A(1,1,1)…

A:

Q: The vector field F(r, y) = (2xy +y) dx + (æ + 2xy) dy is conservative. %3D The function f(x, y) = x²…

A:

Q: Let a vector field F(x,y)=sinx i + xy j be given. What is the integral representing the work done by…

A:

Q: Evaluate F Î ds for the vector field F(x, y) = yì – xj counterclockwise along the circle x² + y² = 1…

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: r(t) (cos t, sin t, t) for over 2. Compute the integral of the vector field F(x, y, z) = (2x, 3y,…

A: We will find out the required value.

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