Consider the closed surface S consisting of the graph z = 1 - x2 - y2 with z 2 0, and also the unit disc in the xy plane. Give this surface an outer normal. Compute ds where F(x, y, z) 3D (5х, 5у, z). F• dS =

Q: Consider the rormal line to the surface z= z + y? (parabokoid) at the point P(1,1, 2) Find the point…

A:

Q: Evaluate the surface integral. || x²yz ds, S is the part of the plane z = 1 + 2x + 3y that lies…

A: Given: ∫∫sx2yzdS where S is the part of the plane z=1+2x+3y that lies above the rectangle…

Q: Let a: ISc R be a curve on a. regular surface S and consider the ruled surface generated by the…

A: Given: α→:I→S⊂R3 α→,N→t

Q: Let S be the surface given by a piece of the plane y = 0, with 0 < x <1 and 0 < z< 2. the unit…

A: Given: To find the flux of F through S by considering F = 3j + k. Let's take unit normal n =j.

Q: 1. Let F(x, y, z) = yi+ xj+ yzk and the surface S be the part of the paraboloid z = 4 – x2 – y? that…

A:

Q: Let C be a simple closed smooth curve in the plane 2x + 2y + z = 2, oriented as shown here. Show…

A: Given, The equation of the curve is 2x+2y+z=2. The closed integral is ∮C2y dx+3zdy-x dz Let…

Q: (a) Prove that M: (x² + y^y° + 3z² = 1 is a surface. (b) For which values of e is M: z(z – 2) + xy =…

A: Consider the provided question, (a)

Q: Consider the surface S= {(x, y,=)| xy + z =5; x20; y20; z20} in R. Use the Method of Lagrange to…

A: Given  surface S=x,y,x   ; xy+z=5   ;   x≥0 , y≥0  , z≥0 consider the point  x,y,z on the surface S…

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: Evaluate || f (x,y,z) dS where S is the portion of the sphere centered at the origin of radius two…

A: we have the equation of sphere x2+y2+z2=22s is the portion bounded between xy plane and z=1 in the…

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: Evaluate the surface integral xyzdS where S is the part of the plane z = 1 + 2x + 3y that lies above…

A:

Q: Consider the normal line to the surface z = x2 + y² (paraboloid) at the point P(1, 1, 2). Find the…

A:

Q: Determine the projection of the following surface on the xy plane. (xy trace) 4z-목 + =0

A: here first solve the given surface and then we trace it

Q: Evaluate the surface integral| f(x.y.z) dS using an explicit representation of the surface. f(x.y.z)…

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: Let (S) be the surface given by the following equation. x²+y² = 1+ z?. An equation of the tangent…

A:

Q: Evaluate the surface integral  ffR(x+y)dS where σ is the portion of the plane z = 6 − 2x − 3y in 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 (S) be the surface given by the following equation. x²+y? = 1+ z?. An equation of the tangent…

A:

Q: Determine the projection of the following surface on the xz plane. (xz trace) 4x – + = 0

A: Given equation of surface is 4x-y24+z28=0

Q: (4) Use Stokes' Theorem to evaluate 11. Vx (yz, xz, xy) dS where S is the surface S that is the part…

A:

Q: a) Evaluate (-ri-yj+2z k) -n dS, where o is the portion of the paraboloid z = 3r + 3y? between the…

A:

Q: b) Use Gauss theorem to evaluate F.ndS, where F(r, y, 2) = (2x²y² + In z) i – xy³ j – tanh(a² + y³)…

A:

Q: Find the area of the part of the paraboloid z=x² + y² that lies under the plane z=9.

A:

Q: 2. Use Stokes' Theorem to evaluate , Vx F ·n dS where F(x, y, z) = and C is the part of the sphere…

A:

Q: Find the surface area of the paraboloid z =1-x² - y? that is above the xy-plane.

A:

Q: Consider the set S given by S = {(x, y, z) ER³ : 2 = 2x² - y²}, where A E R\{0} is constant.…

A: As per the question we are given a quadratic surface as : S = {(x,y,z) ∈ R3 | z =2x2 - λy2} And we…

Q: Let (S) be the surface given by the following equation. x²+y? = 1+ z?. An equation of the tangent…

A:

Q: Let S be the closed surface composed of the parabaloid z = x² + y? and the planes z =2 and z = 4.…

A: Please see the answer below.

Q: Compute the integral of the Gauß curvature over the surface shown in the picture. The three boundary…

A:

Q: The tangent plane to the graph of the surface z=In(x2+y2) at the point (1,0,0) contains point…

A: Given function is, z=lnx2+y2

Q: Consider the surface z = 1? – y?. Let Tp denote the tangent plane at a point P on this surface. Find…

A: Given equation of surface in the form of z=f(x,y) so first write in the more general form of…

Q: Calculate F- dS if F(x, y, z) = x²ye²i+ æy?j – 2xye*k, where S is the surface of the box bounded by…

A:

Q: b) Use Gauss' theorem to evaluate RR F. ndS where F = (x² - y)i + y³z²j + (xy – y²z³)k UTM UTM and o…

A:

Q: Consider the surface given by x2 + y? + z? = a². Show that every normal line to this surface passes…

A:

Q: Let P be the tangent plane to the surface z=2x2+y2 at the point Q=(1,1,3). Which of the following…

A: tangent planes to a surface are planes that just touch the surface at the point and are “parallel”…

Q: Consider the surface: S:z = 4 – a?, in the first octant and bounded by x² +y°, as shown in the…

A: The equation of the surface given is as follows: z=4-x2 The surface is bounded by the curve x2+y2=4.…

Q: Find the surface normal vector to the surface z 4-(x +y²) at the point (1, 1,v2) by (a) Vf (b) N =…

A:

Q: a) Evaluate ds, where the curve C is the intersection between the surface z = V1- y? and the plane r…

A:

Q: Evaluate || F:ñ dS where S is the surface bounded by z = a - Vx² + y² and the xy-plane including the…

A: The vector field is F→=xπ, yπ, 3z-2π. The surface is bounded by z=a-x2+y2 and xy-plane. The function…

Q: Consider the closed surface S consisting of the graph z = 1 - x² – y? with z 2 0, and also the unit…

A: Surface integration

Q: Use Stokes' theorem to calculate (V × A) · ndS with A = 3yi – xzj+ yz²k, where S is the surface of…

A:

Q: Determine the sections of the following surface on planes parallel to the xz plane. -y+ = 1 %3D O…

A: here first we solve for the surface and then check it .

Q: a) Evaluate (-x i - yj+ 2z k) - n dS, where o is the portion of the paraboloid z = 3r2 + 3y? between…

A:

Q: Consider a surface S which is parameterised by {r(u, v) = ui + (u + v)j+ vk :0 < u < 1,0 < v< 1}. At…

A: We have a surface S={r→(u,v) =ui^+(u+v)j^+vk^ :0≤u≤1, 0≤v≤1} Normal vector to the surface S is given…

Q: Graph the surface 1 = y2 + z2 –- by first sketching at least three cross sections in each of the xy…

A: We have to sketch the graph of the following surface in the 3D space: . First we have to sketch the…

Q: (a) Find the surface area of the plane 3x +2y + z = 6 that lies in the first octant.

A: If z=fx, y then surface area is, AS=∫∫D1+zx2+zy2dA

Q: Find the area of the part of the plane 5x + 4y + z = 20 that lies in the first octant. (Use…

A: Part of the plane equation   5x+4y+z=20⇒z=20-5x-4yPlane lies in the first octant , so x≥0 ,…

Q: Evaluate the surface integral. || x2yz ds, s is the part of the plane z = 1+ 2x + 3y that lies above…

A: