Let C be the intersection curve of the paraboloid z = x2 − y2/2 with the cylinder x2 + y2 = 1, traveled counterclockwise when viewed from above (that is, its XY projection travels counterclockwise). Let I = (image 1), with F⃗ (x, y, z) = (3xy/2,3x2/4 + xz, −xy/2). When applying Stokes' Theorem to calculate the line integral I, the resulting double integral corresponds to: (image 2)

Q: Given the scalar functionf (x, y) = 2xy, calculate the line integral along the ellipse curve+= 1…

A: To Evaluate: ∫CF·dr, where F(x,y)=2xy and the curve is  C: x29+y24=1.

Q: Let C be the curve that consists of 4 straight intervals, beginning at (0, 0, 0) , going to (5, 2,…

A: Consider the given: F=8x+6y+3z+yz ,8x+16y+3z+xz,2x+3y+6z+xy and the points:(0,0,0),(5,2,2),(8,5,6)…

Q: Find the coordinates of the centroid of the curve given belov x = cos t+tsin t, y =t cos t- sin t,…

A: Given data: The x-coordinate of the curve is x=cos t+tsin t. The y-coordinate of the curve is y=tcos…

Q: Let F(x, y, z) = 2yzi+ 4.xzj+ e*yk, and let C be the circle x2 + y² = 16, z = 3, oriented…

A:

Q: 11. Use Stokes' Theorem to evaluate OF dr where F=(xz,e",y+z) on the curve that is the intersection…

A: The equation of curl is F=xz,ey,y+z. The equation of plane is 6x+3y+z=12

Q: Compute the flux of F = 3(x + z)i +j+ 3zk through the surface S given by y = x2 + z2, with 0 0, z 2…

A: flux is determined as shown below

Q: Find the line integral with respect to arc length (8x + 9y)ds, where C is the line segment in the…

A:

Q: Integrate f(x,y) = √(64 - x2) over the smaller sector cut from the disk x2 + y2 ≤ = 64 by the rays θ…

A: Concept:   The calculus helps in understanding the changes between values that are related by a…

Q: Use Stokes theorem to evaulate ⌠c F*dr where F(x,y,z)=2i+yj-xk and C is the boundary of the part of…

A: F(x,y,z)=2i+yj-xk C: z=9-x^2-y^2  z=0

Q: Calculate F - dS where F (e* sin y, e² cos y, yz²) S is the surface of the box bounded by the planes…

A:

Q: Find the circulation of F(x, y, z) = (-y sin 2, x sin z, xy cosz +12) around the circle cut from the…

A: Given that,F(x, y, z)=(-ysinz, xsinz, xycosz+12) around the circle cut from the sphere x2+y2+z2=5 by…

Q: Compute the flux of F = 3(x + z)i +j+3zk through the surface S given by y = x2 + z2, with 0 0, z >…

A: From given data we see that the surface is oriented towards xz plane,  Hence,  d→s=dxdzj→

Q: Find the tangent plane to the surface z = cos(3ry) + z' + 2y at the point (1,0). (z = cos(3ry)+z'+…

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Q: Use Stokes' Theorem to caleulate ſ. F - dř, where F = zyi+3=3+5zk, and C is the closed path around a…

A:

Q: Use cylindrical coordinates to evaluate the triple integral □ لاا √² + y²dV, where E is the solid…

A:

Q: Use Stokes' theorem to evaluate frotF. ds. F(x, y, z) = xyi + e²j + xy²k; S consists of the four…

A: To find- Use Stokes' theorem to evaluate ∫SrotF·dS, Fx, y, z = xyi + e2j + xy2k; S consists of the…

Q: Calculate F - dS where F = (e² sin y, e² cos y, yz²) S is the surface of the box bounded by the…

A:

Q: Use Stokes' theorem to calculate JJ (A) nds by A = 3yi-xzi+yz²k where S is the surface of the…

A: The given vector field is: A=3yi-xzj +yz2k The surface S is the paraboloid 2z=x2+y2 bounded by z=2.…

Q: Let C be the curve intersection of the sphere x'+y+=9 and the cylinderx+y° =5 above the xy-plane,…

A: To Evaluate: ∫CF·dr Using Stoke's theorem. where F→=<2yz,5xz,lnz> and C is the curve of the…

Q: 2)Use Stokes' Theorem to evaluate [F.dr where E2xyi, y² j‡ 2xzk and Sis the first octant с surface…

A:

Q: Use Stokes's theorem to evaluate o F· dr, where F(x, y, 2) = xy² i+x²yj+yz k, where C is a…

A:

Q: Compute the flux of F = 3(x + z)i +j+ 3zk through the surface S given by y = x2 + z2, with 0 <y<9, x…

A:

Q: Calculate F- dS where F = (e" sin y, e* cos y, yz²) S is the surface of the box bounded by the…

A:

Q: Find the counterclockwise circulation and the outward flux of thefield F = (-sin y)i + (x cos y)j…

A: Use tangential form of Green's Theorem to find the counterclockwise circulation. Counterclockwise…

Q: Suppose that C is the curve of interesection between the plane z = x and the cylinder x² + y? be the…

A: Consider the given function. z=x and x2+y2=1 As, the function S be the interior of this ellipse…

Q: Find the line integral with respect to arc length / Q = (0,3). (2x + 7y)ds, where C is the line…

A:

Q: Use Stokes' Theorem to evaluate F. dr, where F(x, y, z) = zi + yj – rk and C is the boundary of the…

A:

Q: State what the Divergence Theorem says, using your own words. You may assume that E is a solid in R³…

A: We can state the Divergence theorem .

Q: B. Compute the shared mass between the polar curves (r = 1+ cose) and (r = 1- cose), usę density…

A:

Q: Integrate F =-(y sin z)i + (x sin z)j + (xy cos z)k around the circle cut from the sphere x2 + y2 +…

A:

Q: Calculate F - dS where F = (e" sin y, e" cos y, yz²) S is the surface of the box bounded by the…

A: Given F¯={exsiny,excosy,yz2)

Q: Find a parametrization for the curve y = Vx with terminal point (0, 0) using the angle 0 in the…

A:

Q: Use Stokes' Theorem to evaluate L* + cos x) dx + (sin y + z²) dy + x dz where C is the closed curve…

A: To Evaluate: Use Stoke's Theorem:

Q: Given the scalar functionf (x, y) = 2xy, calculate the line integral along the ellipse curve ²² +²²=…

A: We have to find the line integral along the given curve.

Q: Find the work done by F in moving a particle once counterclockwise around the given curve = (2x –…

A:

Q: Let F(x, y, z) = (8 + 8y+ 3z+yz, 6x + 6y + 2z+ cz, 5x + 4y+ 10z+ ry). and let C be the curve that…

A:

Q: 4. Obtain the equation of the tangent plane to the surface with parametric representation r(u, v) =…

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

Q: 6. Compute the flux of F = [0, 0, z²] through the spherical surface S given as the upper hemisphere…

A: To find- Compute the flux of F→ = 0, 0, z2 through the spherical surface S given as the upper…

Q: Given the scalar functionf (x, y) = 2xy, calculate the line integral along the ellipse curve+ 1 from…

A: Here we have to find the value of the line integral of function f(x,y)=2xy alone the ellipse…

Q: In which of the choices below is a parametrization for the tangent line to the curve of intersection…

A: “Since you have asked multiple question, we will solve the first question for you. If youwant any…

Q: Use the polar coordinates to evaluate volumes : 2) Below z = V1- x2 – y2 inside x² + y2 = above the…

A: Using the polar coordinates we have to evaluate the volumebelow z=1-x2-y2 inside x2+y2=14 above xy…

Q: C is the intersection curve between the paraboloid z = x² + y? and the plane 6x + 8y – z = 21,…

A: Strokes theorem:

Q: Find the flux of F across S, where F(x, y, z) = xyi + y²j + zk and S is the part of the paraboloid z…

A: According to question given that F(x,yz,)=xyi^ + y2 j ^+z k^S is the part of the paraboloid z=x2+y2,…

Q: verify Stokes § = (a%-y)i - yz'i over the uppea hazf SOD Face Of the sphere =1 bounded by the

A: Given that,  F→=(2x-y)i→-yz2j→-y2zk→ To verify Stoke's theorem: ∫CF→·dr→=∬Scurl F→·n^dS Here C is…

Q: Use Stokes’ Theorem to evaluate F(x, y, z) = 3zi + 4x j + 2yk; C is the boundary of the paraboloid…

A: Use Stokes theorem

Q: Let F = (x'e* +2.xy - 2y)i+(cos y+x² +3x)j and let C be the ellipse with equation (2x - 2y) +(x+ y)°…

A:

Q: 2)Use Stokes' Theorem to evaluate | F.dr where F =2xyi- yj+2xzk and S is the first octant surface of…

A: The given problem is to evaluate the line integral in the surface of first octant with upward…

Q: Use Stokes' Theorem to calculate fe F-dĩ, where F = zyi+3=j+5zk, and C is the closed path around a…

A: Here given that F→=xy i^+3z j^+5x k^ and curve C in xz-plane. To find: ∫C F→·dr→ by Stokes' theorem.

Q: Find the line integral with respect to arc length (2x + 6y)ds, where C is the line segment in the…

A: The solution of line integral is given as follows :

Domain

In simple mathematics terms, a domain is defined as the space on which all the possible inputs for the function will lie. It can also be expressed as a set of departures for the required function.