4. Evaluate fo F• dr by the Stoke's theorem, where F = -y? î+x j+ z² k, and C' is the curve of intersection of the plane y + z = 2 and the cylinder x² + y² = 1.

Q: 15. If the surface S has projection R on the xy-plane, show that F.īdS = S da dy ה.F provided |ñ -…

A: Question: If the surface S has projection R on the xy-plane, show that ∬SF¯·n¯dS=∬RF¯·n¯dxdyn¯·k¯…

Q: If (a,ß.r) is a point at which the surface x2+y2-z-2x+42=0 has a horizontal tangent plane, then Iyl=

A: Horizontal tangent plane

Q: 18. Evaluate J (V x F) · N dS where F = yi + (x- 2xz) j- xy k and S is the surface of the sphere x…

A: Stokes theorem is used

Q: 2 Consider the line L: y -1 +t|0, te R, and the plane T : x – 4y – z = 6. (a) Show that L is…

A:

Q: 6 Consider points P and Q in the x-y plane, with P = (1,0) and Q (0,1). The line integral %3D %3D Q…

A: Introduction :Given, two points P and Q .P= (1,0)Q= (0,1)We have asked for the line integral of

Q: If F= 2y I- 3J +x²k and s the surface parabolic cylinder y= 8x in bounded by the planes OF the 2.…

A:

Q: Find an equation for the tangent plane to the surface z = 6x3y2 + 3y at the point P(1, -3). Hint:…

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Q: 25. Evaluate F.N dS where F= yl + 2xj –- zk and S is the surface of the plane 2x +ye 6 in the first…

A:

Q: An equation of the tangent plane to the surface z = 4xy² + 3x²y² at the point (1, 1, 7) is: Select…

A: z= 4xy2+3x2y2F(x,y,z) =…

Q: Calculate F dS if F(x, y, z) = x²yei+ xyj – 2xye k, where S is the surface of the pox bounded by the…

A: Given-F(x,y,z)=x2yezi+xy2j-2xyezk To find- ∬SF.dS ,where S is the surface of the box bounded by the…

Q: Given F(x, y, z) = -yzi + xzj – xyzk be a vector function that passes through a surface S of z + x2…

A:

Q: 8. Let S be the surface {(x, y, 2) | a = y? + 22 + 1}. Find every point (ro, yo, zo) on S so that…

A:

Q: Let f(x, y, z) = x² + 2y? + 4z2. Find the point on the plane x + y + z = 14 at which f has its…

A: Given: fx,y,z=x2+2y2+4z2and the plane equation x+y+z=14 To find the point x,y,z The Lagrange's…

Q: 4. Where does the curve r(t) = ti + t²j+ (3 t2)k intersect the paraboloid z = x2 +y² ?

A:

Q: Evaluate ſ F . n dS if F = (x + y² ™)ī − (2x)j + (2yz)k and $ is the surface of the plane 2x + y +…

A:

Q: Compute x dS where S is the portion of the plane z=1+x+2y that lies over the square [0, 1] × [0, 1].…

A: We have to find the surface integral ∫sxds.

Q: Let G be the solid enclosed by the plane z= y, the xy-plane, and the parabolic cylinder y = 1-x². If…

A: Consider the given information,

Q: Find the sum of maximum and minimum values of the function f(x, y, z) = 6x + 12y+ 18z on the elipse…

A:

Q: If F = 2y i - zj + x²k and S is the surface of the parabolic cylinder y²= &x in the first octant…

A:

Q: If (a,B.y) is a point at which the surface x2+y2-z²-2x+12=0 has a horizontal tangent plane, then y|…

A: As per guidelines, we will solve first question only. Let α,β,γ be the point on the surface…

Q: Let f(x, y) = 2x and C be the line segment on the plane from (0, 0) to (3, 4). Evaluate the line…

A:

Q: Let C be the curve of intersection of the paraboloid x² + y² = z and the cylinder x² + (y − 1)² = 1.…

A: For the given curve we have the following vector valued function: R⇀=2cost. 1+sint,…

Q: 1. Evaluate the line integral f(z, y) ds where f(x, y) = Vx+ 2y and C is the straight-line segment…

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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: 59. Find all points on the portion of the plane x+y+z35 in the first octant at which f(x, y, z) = xy…

A: We have to find all points on the portion of the plane x+y+z=5 in the first octant at which f(x, y,…

Q: F. dS, where F(x, y, z) = z²xi + 2) + (x2z + y2)k and S is the top half of the sphere x² + y² + z? =…

A:

Q: 6. C is the curve of intersection of the plane 4x + 5y+ z = 0 with the cylinder x2 + y? = 4 directed…

A:

Q: (8) Find the point(s) where the line r(t) = ti+(1+t)j+5k intersects the paraboloid z = x² + y?.

A: A detailed solution is given below.

Q: Evaluate the integral [(3x – 2y + z)ds where C is the curve, C, and C, as shown. 3 y²+ z² 2 G 2* 3…

A: Given that ∫c3x-2y+zds. The objective is to find the integral value. ∫c3x-2y+zds=∫c13x-2y+z…

Q: 2. Evaluate ([.(Vx F). ds, where S is the surface of the paraboloid x2 + y2 + z = 4 above the…

A: We have given F→=(x2+y-4)i+3xyj+(2xz+z2)ksurface is given x2+y2+z=4 and above the xy planediagram of…

Q: a) Evaluate [-ri-yj+2: k) - n ds, (-a i- where a is the portion of the paraboloid z = 3r + 3y…

A:

Q: Consider the PDE y u + Uyy + u, - Uy = 2 +y? In what region of the xy-plane is this PDE elliptic? xy…

A: The objective is to find in what region of the xy-plane the pde is elliptic.

Q: Use the Green's Theorem to evaluate the line integral 2xydx + x² – y² dy where C is the curve…

A:

Q: 4. Compute |F- dš, where F = (y, –x,2z) and S is the surface of the sphere of radius 2, centered at…

A:

Q: 9. Show that the sum of the T-, y-, and z- intercepts of any tangent plane to the surface VI + Vỹ +…

A: Given, Gradient of the surface at any point (x0,y0,z0) is given by,

Q: Find an equation of the plane tangent to the surface xy z2 = 2 at the point (2, 1, -1). О А.х+ бу +…

A:

Q: 11. Let D be the solid inside the cone z = √²+ y², inside the sphere x² + y² +2²=9 and above the…

A:

Q: 3. (a) Show that the two surfaces S₁ z = xy and S₂ : 2 = x² - y² intersect perpendicularly at the…

A: Introduction: The gradient of a surface is a vector quantity that is directed in the normal…

Q: 4. Evaluate //F FdS, where F(x, y, z) = (2x + 2xz, -4xy², y - z2) and R R surface of the solid…

A: The given problem is to evaluate the given surface integral over the given solid surface bounded by…

Q: O 9/ Compute I - [.lryz) , uhere s is the part g the plane y +x that lies inside the half-cylinder…

A: We know that, dS=1+∂z∂x2+∂z∂y2dydx where S is the part of plane z=y+x. ∂z∂x=1∂z∂y=1 dS=1+(1)2+(1)2…

Q: 7. Find the flux of F(z,y, 2) = ri+yj+zk through the portion of the paraboloid z = 1+z²+y° that is…

A: Given

Q: where F(x, y, z) = (ry, x²y, y) Use Gauss's theorem to calculate where is the paraboloid z = 4 – x²…

A:

Q: Find the sum of maximum and minimum values of the function f(x, y, z) = 4x + 8y+ 12z on the elipse…

A:

Q: 7. Consider the Surface S:z = 52² + 5y?. An equation of the plane tangent to Sat the point P (3, 3,…

A:

Q: 3.Evaluate || F. ds where F= zi+x j +y k and 'S' is the surface of the cylinder x'+ y=1 contained in…

A: Given that ∬S F→·dS→ where  F→=z i^+x j^+y  k^ .......1and  S is the surface of the cylinder x2+y2=1…

Q: find ⌠⌠F*dS if F(x,y,z)=2xi+2yj+2zk and S is the paraboloid z=8-x^2-y^2 lying above the rectangle…

A:

Q: 17. F = |r|r, where r = xi + yj + z k, S consists of the hemisphere z = /1 – x² – y2 and the disk x?…

A: The solution are next step

Q: Find an equation for the tangent plane to the surface z = 3y^2 − 2x^2 + x at P(2, −1, −3). Express…

A: Let's find equation of tangent plane.

Q: Show that the tangent plane to the surface x 2 a 2 + y 2 b 2 = cz at the point P(x0, y0, z0) is…

A:

Q: Let F(x, y, z) = (x,y,z). Compute ∫∫sF · dS where S is the part of the plane x + y + z = 1 in the…

A: