Apply Stoke's theorem to calculate S. (x + y – 4)dx + (2x – z)dy + (y + z)dz, where C is the boundary of the triangle with vertices (1,0,0), (0,3,0), (0,0,5) oriented in the anticlockwise direction.

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Q: 2. Use Green's Theorem (and only Green's Theorem) to evaluate |(2? + y)dx + (x² – y³)dy, where C is…

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Q: What is the equation of the plane to the surface z = e3y sin(3x) +2 at the point P ,0, 3 %3D O x- y…

A: The solution is given in the next step.

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A: z=13-x2-y2 Partially differentiating with respect to x, we get zx=0-2x-0=-2x Partially…

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

A: see 2nd step

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A: Problem is solved using concept of double integral.

Q: x y 11. Use Green's theorem to evaluate O In(1+ y) dx –- dy , where C is the triangle with vertices…

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Q: Evaluate the surface SS₂ xz ds S is the part of the 2x+2y+z=4 that li- the first octant.

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Q: 4. Use Stokes’ theorem to evaluate fa F. dr, where F(x,y, z) = (xy,yz, zx), and C is the triangle…

A: I am going to solve the given problem by using some simple calculus to get the required result.

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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)

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Q: (e) If E C R³ is the ellipsoid (x/2)2 + (y/3)² + (z/4)² = 1 and F evaluate the integral ffE curl F-…

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Q: Let u(x,y) satisfy the following equation uxx+(x2+y2-1)uyy=0 Find the region in which the equation…

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

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Q: The parametric form of the solutions of the PDES Uy + u Uz = 0 , u(x,0) = x is (A æ (s, t) = st + s,…

A: We need to find the parametric form of the solution of the partial differential equation uy+u ux=0 ,…

Q: Sſcos( dxdy y-x y+x D

A: Given and u=y-x,  v=y+x where D is the trapezoid with vertices (1, 0), (2, 0), (0, 2) and (0, 1).

Q: a) Evaluate x²y? ds, where the curve C is the intersection between the surface z = := V1– y? and the…

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Q: The plane z =0 separates region 1 (z 0), which is a dielectric material with ɛ, = 4, from region 2…

A: The complete solutions are given below

Q: Evaluate (y + Vx) dA, where R is the triangular region with vertices (0, 0), (1,0), and (0, 1). R

A: The solution starts from step 2. The information presented below will be helpful to understand the…

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Q: Verify Green's theorem for Q7 [x(x? + y²)dx – y(3x² – y²)]dy Where C is the boundary of the triangle…

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Q: a) Find the points at which the twisted cubic F(t) = ti + t} +tk intersects the plane 4.x +2y+z = 24…

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Q: Use the flux form of Green's Theorem to evaluate |(6xy + 8y°) dA, where R is the triangle with…

A: Greens' Theorem : Let D be the region bounded by the curve C. Let F(x,y)=P,Q be the vector field…

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

A: As per our guideline, we are supposed to solve only first question. Kindly repost other questions as…

Q: et T(x, y, z)= 100+ x² + y² represent the temperature at each point on the sphere x² + y² + z² = 80.…

A: Given: Tx,y,z=100+x2+y2sphere g= x2+y2+z2=80plane h =x-z=0

Q: . Let C be the triangle with vertices (0,0), (1,0), and (1,1), traversed counter- clockwise.…

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Q: Use Green's Theorem to evaluate | (v? – tan-1 x²)dx + (3xy + 4x – In (y2 + 4))dy where C is the…

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Q: 3. Find the area of the band cut from the paraboloid x² + y? – z = 0 by the planes z = 2 and z = 6.…

A: Let's find.

Q: Let F = < 4x + y², 5y + z, 9z + x So F. dī, Use Stokes' Theorem to evaluate C where C is the…

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Q: f(x, y) = xy over the triangular region with vertices (0, 0), (8, 0), and (0, 6) O 12 2 96 O 16

A: Solve

Q: Green’s Theorem for line integrals Use either form of Green’sTheorem to evaluate the following line…

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Q: 10. Using Stoke's theorem, evaluate y dx+ xz dy+x² dz where C is the boundary of the triangle cut…

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Q: Evaluate the surface integral.

A: We need to find the surface integral of

Q: 3. Let H = 2xi – 3xy j+xz² k. Calculate ſ, Ĥ •d where S is the surface of the cube with corners at…

A: Given- Let H→ = 2xi→ - 3xyj→ + xz2k→ To find- Calculate ∫SH→·dA→ where S is the surface of the cube…

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

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Q: Consider the surface S: z = 4x2 + 2y2 . An equation of the plane tangent to S at the point P (−1, 2,…

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

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

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Q: Ex.5 Given the Jordan curve y that is the counterclockwise oriented boundary of the set T= {z=x+iy :…

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Q: Use Stokes's Theorem to evaluate EF. dr. In this case, C is oriented counterclockwise as viewed from…

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Q: (a) The part of the surface z = 1+ 3x + 2y² that lies above the triangle with vertices (0,0), (0,…

A: a) Given part of surface is  z=1+3x+2y2, which lies above the triangle with vertices (0, 0), (0, 1),…

Q: Let h, r be constant. Consider the quadric surface S of equation: (x − h)2/9 − (y − 1)2/3 + (z −…

A: Substitute y = 5 in the given equation of quadratic surface S then simplify to know the resultant…

Q: Use Cartesian coordinates to evaluate y² dV D where D is the tetrahedron in the first octant bounded…

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Q: Find any points on the hyperboloid  x2 − y2 − z2 = 6  where the tangent plane is parallel to the…

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Q: (b) A = 2rza, + 3z sin ø a, – 4 rcos o a̟ and S is the surface of the wedge 0 <r< 2, 0 < 0 < 45°, 0…

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Q: (x - y) dv, where E is enclosed by the surfaces z = x² - 1, z = 1 - x², y = 0, and y = 4

A: Consider the given function fx,y,z=x-y.  The region E is enclosed by the surfaces…

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

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