20. Which of the following vectors is normal to the surface In(x + y* -- z) = x-1 at the point where y-8 and z = 4 A) i-j-2k B) 2i-3j+k C) i+2j D) -2i+ j+3k E) j-3k

Q: Let C be the positively oriented square with vertices (0,0), (1,0), (1, 1), (0, 1). Use Green's…

A: Let's solve by using green theorem.

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: 2. Prove that fo r*dr + xydy = -+- where C2 is the line segment from (1,0) to (0, 1) by using…

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Q: Let S be the portion of the cylinder y = ln x in the first octant whose projection parallel to the…

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Q: Let Li be the straight line segment from (0.0,1) to (1,a, 8) and let F: (z+ax, ay, x> Calculate the…

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Q: 3. Compute the scalar line integral 3 y – xz ds where C is the line seg- ment from (0, 1, 1) to…

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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: What is the normal vector, n for S is a surface of x +y² +z² = 36 in the first octant? %3D

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Q: The value of the triple integral 6zdV where E is the upper half E of the sphere of x² + y? + z? = 1…

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Q: Verify Green's theorem in a plane with respect to S. [(x2 - y²)dx + 2xy dy], where C is the boundary…

A: Using greens theorem

Q: Evaluate the line integral x'z ds, where C is the line segment from (0, 5, 2) to (4, 6, 5).

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Q: a) Evaluate // (-xi- yj+2z k) · n dS, where o is the portion of the paraboloid z = 3x2 + 3y² between…

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Q: Find a unit vector normal to the surface x^3- xyz + z^3 = 1 , at the point (1, 1, 1)

A: To find the unit normal vector to the surface x3-xyz+z3=1 at the point 1, 1, 1

Q: Evaluate ze*²+y²+z? dV, where E is the portion of the ball centered at the origin of radius 5, x2 +…

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Q: What is the value of F.ds, where F = 4xz,- y° i, + yz, ? Here, s is the surface bounded by x = 0, x…

A: Here function is given find f.ds as below

Q: Give a parametrization, including the appropriate u- and v-bounds, for the portion a cylinder of…

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Q: 18. The volume of the solid that lies between the paraboloid z = 2x² + 2y² and the plane z = 6

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Q: Let d represent the distance from the center of the sphere 22 +y ++ 4x- 10y = 12 to the plane y = 4.…

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Q: S how that the line af intesection of the Planes and 5x-2y-Z=0 X+2y-22= 5 is Paralkel to thelime メ+3…

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Q: Evaluate the line integral / x°z ds, where C is the line segment from (0, 2,4) to

A: Line integral: - The function to be integrated is determined along a curve in the coordinate system…

Q: 2). Find the flux of F(x, y, z) = (0, yz, z) outward through the surface S cut from the cylinder y +…

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

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Q: Calculate ff. (VxF). n dS lf F = (x + 2y)i – 3zj + xk and S is a plane surface 2x + y + 2z = 6…

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Q: (9) Evaluate the surface integral / S, czdS, where o is the part of the plane r+2y+3z = 6 that lies…

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Q: 7. Which surface is parameterized by ř(u, v) = (u, uv, v) for –1 < u <1 and –1< v< 1?

A: r→(u, v) = u, uv, v for -1≤u≤1 and -1≤v≤1The parametric curve is  Option C is correct.

Q: What is the value of J]F ds, ds, where F = 4xzi,- y² % + yzi, ? Here, s is the surface bounded by x…

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Q: Q2// A surface S is defined by x2 + z = 9 and the bounded by the planes y = 0, y = 1,x = 0,z = 0 in…

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Q: Q3// A surface S is defined by 2x2 + z 1 and the bounded by the planes y = 0,y 2, x = 0,z = 0 in the…

A: A surface S is defined by 2x2+z=1 which is bounded by the planes y=0, y=2, x=0, z=0 in the first…

Q: Calculate . (VxF). n dS if F = (x + 2y)i – 3zj + xk and S is a 2x+y+2z=6 plane surface bounded by x…

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Q: b) Use Gauss theorem to evaluate F.ndS, where F(r, y, z) = (2x²y² + lIn 2) i – ry° j – tanh(x² + y³)…

A: Given function  F(x, y, z) = (2x2y2+lnz)i -xy3j -tanh(x2+y2)k we have to find ∬σF.nds by using guass…

Q: 2 The values of a and b such that the surfaces ax - byz = (a + 2)x and 4x2y + z3 = 4 cut…

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Q: Find a vector parametrization for the intersection of the plane x + y+z = 3 and the cylinder x² + z2…

A: solution given below..

Q: Let C be the positively oriented square with vertices (0, 0), (3, 0), (3, 3), (0, 3). Use Green's…

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Q: If E is the solid shape bounded by x 20,y 20,z 20 and the plane 6x + 4y + 3z = 12 evaluate %3D (y +…

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Q: Let C be the positively oriented square with vertices (0, 0), (2,0). (2, 2). (0, 2). Use Green's…

A: Green's Theorem:  For a positively oriented curve C∫CP dx+Q dy=∫∫D∂Q∂x-∂P∂y dAfor the region D…

Q: Which of the following is a parametrization of the line through P = (3, 10, 8) perpendicular to the…

A: Use Parametrization to solve

Q: Find a vector normal to the surface x2 + yz = 5 at (2, 1, 1).

A:

Q: Let C be the positively oriented square with vertices (0,0), (3,0), (3,3), (0,3). Use Green's…

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Q: A parametrization of the portion of a surface z=x²+y2 below the plane z= 16 is given by 7(u,v)=…

A: Since you have sent more than one question so we can just answer one of them because of our…

Q: 2. Calculate the centroid of the thin plate R which bounded by r = 1, y = 0 and, 2² + y² = 4.

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Q: Evaluate the line integral x²z ds, where C is the line segment from (0,6, 2) to (8, 5, 1).

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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: 5. Compute the flux of F(r, y.2) =zi- 2yj+ 2ak across the surface of the cylinder bounded by a+y =9…

A: Solve the problem as shown below.

Q: Evaluate the line integral ∫Cx^3z ds, where C is the line segment from (0,3,8) to (8,4,6).

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Q: Let B be the ball with center at the origin and radius 1 SIS 4(z² + y² + z*)dv = B

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Q: 2. Suppose that S is a surface in R3 with unit normal vector N surface area of S is given by (n1,…

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Q: 8. Let S be the part of the sphere r² + y² + z² = 2 where y 2 1, oriented away from the origin. (a)…

A: Given x2+y2+z2=2 where y≥1

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: 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: OFind the orthogonal Tradectories of the families of y2-5ax 42 - %3D where a being a Parameto

A: For find the orthogonal trajectory replacement of dy/dx with (-dx/dy) is required.