1. Let S be a level set surface of a C' function f : R" → R. (la) Prove if p ES is an interior point of S then Vf(p)=0. (1b) Conclude that the interior of the unit sphere S"-1 centred at the origin in R" is empty. (Revised 2020-11-16)

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: Q19// A scalar filed V = xyz exists over a surface S defined by %3D x2 + y2 9 bounded by the plane…

A: given scalar field V=xyz exists over a surface defined by x2+y2=9 bounded by the plane x=0,y=0, z=0…

Q: 16. Let C be a simple closed smooth curve that lies in the plane x + y + z = 1. Show that the line…

A:

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: Use Green's Theorem to compute y dx+(x³+3xy²) dy where C is the closed curve around the triangle…

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: 1. Let S C R be the surface defined for all 0 < x < 1, -1 < y < 1 by z = r² + y. Compute ffsa ds

A: It can be solved by partial differentiation and integration

Q: 5. (a) Evaluate the surface integral /3² dS where o is part of the cone z = VT + y² that lies…

A: As you post multiple questions According to the guideline I only solve the first problem please post…

Q: Use Stoke’s Theorem to evaluate the line integral ∮C(y+2z)dx +(x+2z)dy +(x+2y)dz, where C is the…

A:

Q: Let D ⊂R^2 be a disk containing the origin and assume that g : D → R is a function given by g(x,y) =…

A: This is a problem of multi-variable calculus.

Q: 28. Show that the circulation of F(x, y, z) = (x², y², z(x² + y²)) around any curve C on the surface…

A: Given figure Stoke's Theorem Here take single integral over closed path C and double integral over…

Q: 6. Evaluate the line integral g. x²y² dx + xy dy where C consists of the arc of the parabola y = x²…

A: (a) Direct method The given line integral is ∮C x2 y2 dx + xy dy. The curve C consists of    C1 :…

Q: 10. Evaluate the integral cz(z-1) where C is a simple closed curve in the positive direction that…

A:

Q: 3. Evaluate if G is a solid in the first octant bounded by the plane y+ z = 2 and the surface y = 1–…

A:

Q: Let H be the segment of the hyperbola x-y°= | Connecting (1,0) to (2, J3). Evaluate the line,…

A: Given  H be the segment of hyperbola x2-y2=1 connecting the points (1,0) to (2,3)  we have to…

Q: |Use Green's Theorem to evaluate | -x'ydx+xy°dy where C is the boundary of the C circle x + y =16…

A:

Q: 15- Evaluate the surface integral f, xy dswhere x is the part of the plaue y = 1-x-z that lies in…

A:

Q: 1) Find the outward flux of F across the region D where F= (6x² + 2xy, 2y + x²z, 4x² y³ ) and the…

A: The vector field is,F¯=6x2+2xy, 2y+x2z, 4x2y3The region D is cut from the first octant by the…

Q: 2. Evaluate f, z³ /x² + y² + z² dV where E is the solid hemisphere that lies above the xy-plane and…

A: Solution:- equation of sphere having centre origin (0,0) and radius=1 x2+y2+z2=1…

Q: Which one of the following is the parametric equa- tions of the line that is normal to the surface…

A:

Q: Suppose f(x, y) satisfies the basic existence and uniqueness theorem in some rectangular region Rof…

A: Consider the given: Let the function f(x,y) satisfies the basic existence and uniqueness theorem in…

Q: Find the plane tangent to the given surfaces at the specific point f(x, y) = x2 ln(2x+y) in (-1, 3)

A:

Q: 1. Use the Divergence Theorem to evaluate /| F.dš where F.dSv F = yx? i+ (xy? - 324) j+ (x³ + y?) k…

A:

Q: EXERCISES 2.10 1. Use Stokes' Theorem to evaluate F.dr for E = (xz,-y,x²y), where C is the boundary…

A: Applying stokes theorem we solve it

Q: (b) Give an example of a parametrisation of the unit circle, C = {(x, y) E R² | x² + y² = 1}, that…

A: The parameterization of a circle  that generates the clockwise orientation of C is given by: x=r…

Q: A surface x(y, z) is defined implicitly xy + yz +x'z' = 3-xZ Calculate ду

A:

Q: Verify Stoke 's Theorem for F = (x² +y – 4) î + 3 xyj + (2 xz + z²) k over the surface of hemisphere…

A:

Q: 4. Use Green's Theorem of the Plane to evaluate the following closed path integral. xy2 dx+ 2x2y dy…

A: According to green’s theorem closed path integral over the curve C is calculated by,

Q: 5. Compute / | F. aš, where F = (y, -x, 2z) and S is the surface of the sphere of radius 2, centered…

A:

Q: Q7. Use Green's theorem to evaluate fy² dx + 3xy dy, where c is the boundary of the semi annular…

A:

Q: Let C be the closed, piecewise smooth curve formed by traveling in straight lines between the points…

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: The surface area of the surface generated by revolving y= 1-x2 for 0<x<1 about the x - axis is S = […

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: (b) Consider F(x, y) = -i+ j, where (x, y) # (0, 0). x²+y2 x2+y2- Without using Green's Theorem,…

A: Consider the provided function,

Q: Q7. Use Green's theorem to evaluate y² dx + 3xy dy, where c is the boundary o the semi annular…

A:

Q: Let Ci be the rectangle with vertices (0,0), (2,0), (0,1), and (2,1) oriented counterclockwise, and…

A: Let C₁ be the rectangle with vertices (0,0). (2,0), (0, 1), and (2,1) oriented counterclockwise, and…

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 a parametrization of the curve of intersection of the surfaces z = x² – y² and z = x² + xy + 1.

A: The given curves are z=x2-y2 and z=x2+xy+1 To find the intersecting curve : Let us equate both the…

Q: 29. Let C be a simple closed smooth curve in the plane 2x + 2y + z = 2, oriented as shown here. Show…

A: Given:

Q: Use Green's theorem to calculate the work done by force F on a particle that is moving…

A: GivenF(x,y)=x32-8yi+5x+8yj,C: Boundary of a triangle with vertices (0,0), (5,0) and (0,5)To…

Q: Let C be a curve given by the intersection of the surfaces z = x2 +y2;z = 3−2x . The value of the…

A:

Q: Let S be part of the outward-oriented hyperboloid of one sheet x² + y² = z² = 1 that lies within the…

A: Given x2+y2-z2=1 and x2+y2+z2=3

Q: 1. Let H be the hyperbolic paraboloid given by z = r² - y, and let P be the plane given by n x = 1,…

A: Given: H be the hyperbolic paraboloid z=x2-y2   and the plane P defined as n.x=1 where…

Q: 1. Evaluate the surface integral SL xz² ds ; o is the part of the cone z = Vx2 + y² that lies…

A:

Q: Evaluate / (7xi – zk) · ndA over the sphere S : x² + y² + z² = | S a) by divergence theorem, (b)…

A: a) By divergence theorem, ∬SA. ndA=∫∫∫V∇.A dV Now,  ∇.A=∂7x∂x+∂0∂y+∂-z∂z=7-1=6 By changing to…

Q: Find the points on the surface xy – z2 = 1 that are closed to the origin.

A: Given surface is xy-z2=1. let xy-z2-1=0⇒z2-xy+1=0 Now distance from origin is…

Q: Covider that te curve c is an inters ecdios of the surfaces SMown below If Cirlt)= (x(t), (t), Zlt)…

A:

Q: Let C CR be the curve from (0,-7,0) to (-2,-3,-8) along the intersection of the surfaces x- y 7 and…

A: Please find the answer in next step

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

A:

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.