2. Evaluate ([.(Vx F). ds, where S is the surface of the paraboloid x2 + y2 + z = 4 above the xy-plane and F = (x? + y – 4) î + 3 xyý + (2 xz + z?) k. Ans. – 4 n

Q: z dv, where E is enclosed by the paraboloid z = x2 + y² and the plane z = 4. Evaluate

A: see 2nd step

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

A:

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

A:

Q: IF F= 3yI - Xy I+ 9ZK and s is the Surface of the Pasabolied 2X =x2 fy Bounded By Evaluate %3D I=2…

A:

Q: Find the surface area of the part of the plane 2x+2y +z=4 that lies inside the cylinder x²+y² = 2.…

A: Given that, the equation of the plane is 2x+2y+z=4 and the equation of the cylinder x2+y2=2. Note…

Q: Evaluate the surface integral |[(x² + y° \dS_where S is the surface of the paraboloid z = x² + y²,…

A:

Q: The attached figure shows the surface S1: y + z= 2, limited in the first octant with surface S2: x =…

A: The area of a surface z=f(x,y) bounded by the region R is given by the integral expression:-…

Q: 3. Evaluate , (x + z)dS where the surface S is the first-octant portion of the cylinder y² + z² = 9…

A: Step:- 1 Since, we know Surface integrals ∫∫Sf(x,y,z)dS=∫∫Df(x,θ) rx×rθ dxdθ…

Q: Consider the solid E determined by the inequalities x² + y² + z≤ 4, and let S denote the surface…

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

Q: Evaluate the surface integral. z2 S is the part of the paraboloid x = y2 + z² given by 0 sx 5 4 1 V5…

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: The value of . (x² +y²) dS, where S is the surface of the paraboloid z = 2- (x² + y2 ) above the x-y…

A:

Q: 2. Evaluate the triple in tegral where E is the y-2x2+2z20nd the plane y= 8. solid bounded by SI…

A:

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: Evaluate ez dV where E is enclosed by the paraboloid z = 4 + x? + y², the cylinder x? + y? = 1, and…

A:

Q: Use the Divergence Theorem to evaluate (5x + 10y + z) ds where S is the sphere x2 + y2 + z2 = 1.

A: In the given problem, S is the sphere x2+y2+z2=1Divergence Theorem:∬SF·n dS=∭Ediv F dVNow, the…

Q: Consider the PDE Y Urr + x Uyy + Ux – Uy x² + y? In what region of the xy- plane is this PDE…

A: compare the given equation with Auxx+Buxy+Cuyy+Dux+Euy+Fu=G A=y,B=0,C=x,D=1,E=-1 calculate B2-4AC…

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: Given a surface 2xy - In (3z- 2) = 10, wherez=f(x, y). Find zx at y =7, z= 1. Leave your answer…

A:

Q: For the surface described by z= - 20y+x² – 10x +3y-2xy | a. At what point are both zx=0 and zy=0 %3D…

A:

Q: 9. Find the surface area of the part of the paraboloid a = 2 – 2? – y? that lies inside the surface…

A:

Q: 9. If F = (x – z) î + (x³ + yz)ĵ + 3xy² k and S is the surface of the cone z = a – Va? +y?) above…

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: 3. Find the volume under the paraboloid 3z = 2.12 + 2y2, above the triangle enclosed by the lines y…

A: First we have to draw graph and find limit according to which we have taken strip along y . Find…

Q: Evaluate .n dS and .ndS if F = (x + 2y)i– 3zj +xk, , and S is the surface of 2x + y + 2z = 6 bounded…

A:

Q: Consider D to be the region below the paraboloid z = 3+x² + y² , inside the cylinder x + y² =1 and…

A:

Q: 11. Evaluate ([(y+z³ )dS where S is the part of the paraboloid x = 4- y – z² that lies in front of…

A:

Q: (a) Compute the area of the portion of the cone x2 + y2 = z? with z 2 0 that is inside the sphere x2…

A:

Q: 1) Consider the solid Q in the first octant, generat surfaces: z = (y – 3)² + 1, 2r + 4z = 16, 1= 1,…

A: Volume of solid: Let z=f(x, y) and z=g(x, y) be the functions of x and y of a solid, and let x=a to…

Q: 10/ Compute e Fids, where S is the surface z =N16-x²-y² oriented up ward and F = <y,-x, zZ %3D 工=?

A:

Q: Find the area of the cap cut from the paraboloid y2 + z2 = 3x by the plane x = 1.

A:

Q: É = ( 2x3 + y°, y3 + z³, 3y²z), S is the surface of the solid bounded by the paraboloid z = 1– x2 –…

A: We will find out the required flux.

Q: Evaluate ∬S(x2+y2)dS, where S is the surface z=4−x, 0≤x≤2, 0≤y≤2.

A: Given z=4-x , 0≤x≤2 ,0≤y≤2 to evaluate ∬Sx2+y2dS where S is surface

Q: Evaluate /F. .7 dS where F = (³+y²z)7+(y²+a²y)7+(2x²z+y^) k" and S is the surface of the cylinder x2…

A:

Q: Find the points on the surface xy + yz + zx - x- z =0 where the ant pla

A:

Q: Let C be the intersection of the surface x² + y² + z² cylinder x² + 3y² = 4, z > 0. = 9 and the

A: As per the question we are given the equation of a sphere and a cylinder (with elliptical cross…

Q: Which of the following is the least distance from the origin to the plane 2x + 2y + z = 1? Lütfen…

A: We have to find

Q: Evaluate e dV where E is enclosed by the paraboloid z = 4 + x² + y², the %3D cylinder x? + y² = 3,…

A: We need to evaluate integral.

Q: 11. Evaluate ([(y² +z? )dS where S is the part of the paraboloid x=4-y -? that lies in front of the…

A: We will evaluate the given integral.

Q: (a) Given that F(x,y,z) = (3x² + 3y²)i – 2xyj + yzk, evaluate S, F·dr for which C is the boundary of…

A:

Q: Verify Stokes' Theorem for F = xyî + yzj + xzk and where s is the surface of the paraboloid z = x² +…

A:

Q: Evaluate || 4.nds , where A=zi+xj – 3y°zk and S is the surface of the cylinder x² + y =4 included in…

A: Let ∅=x2+y2-4  Then the normal to the surface S is ∇∅=2xi+2yj The unit normal is…

Q: a) Evaluate the surface integral ſ 2x²y dS over the surface y² + z² = 1 betwee x = -1 and x = 5.

A: a We have to evaluate the surface integral ∬σ2x2ydSover the surface y2+z2=1 between x=-1 and x=5.…

Q: calculate ∫ ∫ ∫Ty dv, being the T region inside the paraboloid z= 1/(x2+y2)the sphere x2+y2+z2=5

A: We need to calculate the triple integral ∫ ∫ ∫Ty dv, where T being the region inside the paraboloid…

Q: Find a parametrization for the intersection of the surfaces: z = x2 + y2 and  x2 + 4y2 = 1

A: Find the parametric representation for the intersection of the given surfaces.

Ratios

A ratio is a comparison between two numbers of the same kind. It represents how many times one number contains another. It also represents how small or large one number is compared to the other.

Trigonometric Ratios

Trigonometric ratios give values of trigonometric functions. It always deals with triangles that have one angle measuring 90 degrees. These triangles are right-angled. We take the ratio of sides of these triangles.