Find the potential function for the vector field given as 8. F(x, y, z) =< 4x²y³z² + 5y² , 4x²y²z² + 10xy – 7z,x*y*z – 7y > 4 O a. $(x, y, z) = x³y³z² + 5y²x – 7yz %3D O b. $(x, y, z) = x³y³z² + 5y²x – 7xy 4 %3D C. 4 $(x, y, z) = x²y²z² + 5y²x – 7yz %3D O d. $(x, y, z) = -x³y³z² + 5yx² – 7yz 4

Q: The gradient of a scalar field p(x, y) = 4 a4 y6 +3 a4 +2 y4 is given by 16 3 y6 + 12 æ³ 24 a4 y5 +…

A: Given, ϕx,y=4x4y6+3x4+2y4 and F=∇ϕx,y=16x3y6+12x324x4y5+8y3 a) Given integral,…

Q: Find the flux of the vector field F across the surface S in the indicated direction. F = 3x2j -z2 k;…

A: F = 3x2 j -z2 k,y = 2x2,0≤z≤2,-1≤x≤1.

Q: The vector field F = ( , ,2²) has a scalar potential in the region R= {(r, y, z) E R³ : x> 0}

A: ANSWER: TRUE

Q: Find a flux of the vector field ä(M)= (2z– y)i +6xj + yk through a part of the surface S: x - 4y+2z…

A: Given, a→M=2z-yi^ +6x j^ +y k^ Also given surface is, S: x-4y+2z-4=0 So the normal vector to the…

Q: Find the potential function for the following vector field F(x, y, z) = (2xy – 29) î + (x² + z²)j+…

A: To find the potential function, f, integrate each partial derivative. Integrate ∫fxdx=∫(2xy-29)dx,…

Q: Consider the gradient vector field V = Vg of the function g(x, y) = x² + y² + e*²-y´ cos(2xy).…

A: We will use the Divergence Theorem in 2D to solve the given flux integral.

Q: ind the flux of the vector field F = yzi + 7yj + yk over the sphere x2 + y2 + z2 = 4.

A:

Q: Determine whether the vector field is conservative. F(x, y) = xe+²2y (2y + xj) conservative O not…

A:

Q: The conservative vector field: F = (2ry- z, r + 2z, 2y – 2rz). has potential function O 0 - 2y 212…

A:

Q: a) Is the vector field F = (a2 - yr, y?- xy) path independent (conservative)? %3D O True O False b)…

A:

Q: 8. Given the vector field G(x, y, z) = e>+2z (1 + x j + 2xk) : a) Show G(x, y, z) is conservative…

A:

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

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: Find the potential function f for the CONservative vector field 言こ<2xe' x2+y? 2ge %3D fCx,y) e 2.…

A: The complete question is solved in next step

Q: b) Find the flux of the vector field F(x,y,z) = 3xi – yj + 2zk; across o where o is the portion of…

A: The given question belonging to the vector calculus  in which we have to find the value flux of the…

Q: Find the flux of the vector field F(x, y, z) = (yz, y°, xz) %3D across the surface S, where S is the…

A: Given

Q: Let W be the work done by the forced field F(x,y,z) =yi+zj in order to move a particle along the…

A: The formula for work done is W=∫F.dr.

Q: (c) Determine the flux of the vector field F = x³î – 2yĵ + 7z²k across a cone z = Jx2 + y2 + 3 that…

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: Calculate the flux of the vector field (x, y³), out of the annular region between the x? + y? = 1…

A:

Q: Select the irrotational vector field. O A yz î-2xz + xyz k O B. yz î + xz î + xz R OCzî + z2 + yz R…

A:

Q: 2. Given F(x, y) = (4x³y² - 2xy³)i + (2x¹y − 3x²y² + 4y³)j is a - conservative vector field. a. Find…

A: To find the potential function for the given conservative vector field :-

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

A:

Q: Evaluate the outward flux of the vector field F = (2x? – 5e" + sin(2²), –4ry + ln(x*), 5z – 3x²y°)…

A: Given

Q: Find the potential function for the vector field given as 2 F(x.y.?) = Oa. $(x, y, z) =x*y + x²2?…

A: Answer:   The correct option is B.

Q: The vector field F = (, i, 2²) has a scalar potential in the region R= {(x, y, 2) E R³ : r² + y? >…

A: We find out true or false.

Q: X(t) 3t, 6. Determine whether the vector field is conservative and, if so, find a potential…

A: Given Fx ,y =6xy i + 3x2+6y2j

Q: Find the flux in the upward z direction of the vector field F(x, y, z) = (xz³, yz³, (x² + y²)²)…

A:

Q: (b) Solve the electric flux through a paraboloid z = x? +y² – 16 that lies below plane %3D z= 0. It…

A:

Q: 4) Consider the conservative vector field given by: F (x, y) = (y² – ye“, 2xy – e*) A potential…

A: Given vector field is F→x,y=y2−yex,2xy−ex. We have to find the potential function that generates the…

Q: Given the vector field F (x, y, z) = 2xyi + (x² + z?) j +2yzk a. Verify that the vector field…

A: Ans :A)

Q: Consider the vector field F (xr, y) = (2y - 12r*y", 4ry – 9r*y?) is conservative. (a) Show that F…

A:

Q: Find the flux of the vector field F = (y, – z, x) across the part of the plane z = 3+ 4x + 2y above

A:

Q: Find the gradient vector fields for the following potential functions (a) f(x,y) = x²y – xy?

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: A potential function for the vector field F(x, y, z) = (yz, xz + z, xy + y+ 1) is (x. y, z) = xyz +…

A: We’ll answer the first question since the exact one wasn’t specified. Please submit a new question…

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

A: The path integral ∫OAF·dr involves the integration over a specific path, where F is a vector-valued…

Q: If F = (4xy – z³) î + 2x² ĵ – 3xz² k is a conservative force field, then scalar potential of F is…

A:

Q: What is the value of a so that the vector field F(x.y)=(y eX+2xy4) i+(ax2y3+ e%) j is conservative?…

A:

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

A:

Q: Find the potential function for the following vector field F(x, y, z) = (2x cos y – 2z3) î + (29+…

A: F = &lt;2xcosy-2z3, 29+2ye2-x2siny, y2e2-6xz&gt; To check F¯ is conservative  not we have t check…

Q: What is the value of a so that the vector field F(x.y)-(y eX+2xy4) i+(ax²y3+ e%)j is conservative? O…

A:

Q: Given the vector field F(x, y, z) = (2z3 – 4y2 + axy,x² - 8xy – bz, 6xz? –- ay + 4), compute the…

A:

Q: What is the value of a so that the vector field F(xy)=(In y+2xy³) i+(ax3y2+ x/y) j is conservative?…

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: For the force field F = (2xe¬Y, (2y – x²e¬y)) in some open connected region containing P(1, 0) and…

A: Given that, F=2xe-y, 2y-x2e-y If F is conservative then ∂2F∂x∂y-∂2F∂y∂x=0

Q: 6) Given a hemisphere bounded by S,: z = /25 – x2 – y? and S2:z = 0.Find the outward flux of the…

A: The flux of a vector-valued function F(x,y,z)=F1i+F2j+F3k over a surface S is defined as ∫SFdS.…

Q: Show that the vector field F(x,y,z)=yz^2i+xz^2j+2xyzk is conservative, find a potential function f,…

A: The given vector function is F(x,y,z)=yz2i+xz2j+2xyzk The given vector field F is conservative if…

Q: Find the potential function p, for the conservative vector field F(x, y, z) = (2xe? – y²sin x)î +…

A: