Use the flux form of Green's Theorem to evaluate |(6xy + 8y°) dA, where R is the triangle with vertices (0,0), (1,0), and (0,1). R |(6xy + 8y) dA = R

Q: Let F(x, y) = x2 + x² + y2J- (a) Show that F. dr +| F. dr if C and C2 are the semicircular paths…

A: The function given to us is  F(x,y)=yx2+y2i-xx2+y2j

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

A: see 2nd step

Q: Find a parametrization of the line passing through (2, 0, 4) and (4, 1, 2). r(t) =

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Q: Give a parametric representation for the path followed by a heat seeking particle placed at point P…

A: The given temperature field is T(x,y)=64-x2-y3------(1) P(4,2) is the given point Then the gradient…

Q: Find a piecewise smooth parametrization of the path C. (ti + tj 0 sts 1 r(t) = 1st< 2 y = VI (1, 1)…

A: C:y=√x

Q: (a) Transform the metric ds? = xydx² + x³y°dy in polar coordinates. %3D

A: We’ll answer the first question since the exact one wasn’t specified. Please submit a new question…

Q: An equation of the tangent plane to the surface z = 4xy² + 3x²y² at the point (1, 1, 7) is: Select…

A: z= 4xy2+3x2y2F(x,y,z) =…

Q: Given F(x, y, z) = -yzi + xzj – xyzk be a vector function that passes through a surface S of z + x2…

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Q: Let G be the solid enclosed by the plane z= y, the xy-plane, and the parabolic cylinder y = 1-x². If…

A: Consider the given information,

Q: The line of action of the 4800-lb force F runs throu the points A and B as shown in the figure.…

A: The scalar component a→ along with b→ will be:                     Compb→a→=a→.b→b→…

Q: Study the analytically of w = L_r(z) at the points z = 0, z = v2,z = -2i

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Q: Find a parametrization of the paraboloid given by f (x, y) = x 2 + y2.

A: To find a parameterization of the paraboloid given by: fx,y=x2+y2

Q: Use Stokes's Theorem to evaluate JeF - dr- . In each case, C is oriented counterclockwise as viewed…

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Q: Use Gauss's Law to find the charge enclosed by the cube with vertices (+/-1, +/-1, +/-1) if the…

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Q: The scalar component of H parallel to the line y=1, z=2 where H = -2a, + 3as- 4a, at p (1,90, 2) -3…

A: Given: H=-2aρ+3aϕ-4az p=1,90,2

Q: Find of and Duf to :- f cas,z) = In (x²+zy?+32) atp(-v34) in the direction of ã=-3i-4j-12K. -314…

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Q: find the outward flux of F across the boundary of D.  F = 2xyi + 2yzj + 2xzk D: The cube cut from…

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Q: (a) Evaluate f. (sin x + 3y²)dx + (2x – e-y*)dy, where C is the boundary of the half-disk x² + y? <…

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Q: 4. Evaluate fo F• dr by the Stoke's theorem, where F = -y? î+x j+ z² k, and C' is the curve of…

A: Given :

Q: Wrons kiant y" + p(+) y # q (4) j'arlt)g =0 +q(4) j' +r (t)y=0 solotions of fifferential equations…

A: Determinant properties

Q: Find a piecewise smooth parametrization of the path C. 0 st<1 (ti+tj r(t) = %3D 1<t<2 (1, 1) y=x

A: Given rt=ti + tj                             0≤t≤1                                       1≤t≤2

Q: G' are centroids of the two triangles ABC änd A' B'C! Show that AA' + BB' t CC =3GG' %3D

A: If G and G` be the centroids of the triangle ABC and triangle A`B`C` respectively then show that,…

Q: Use Gauss's law to find the charge enclosed by the cube with vertices (±1,±1,±1) if the electric…

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Q: a) Evaluate f, (2x – z) dS, where S is part of the plane 3x + 2y + z = 1 that lies above the…

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Q: Let z = g(x, y) = f(3 cos(cy), y + ey) provided that f(3,5) = 8, fi(3,5) = 2, f2(3, 5) = 3. i) Find…

A: Chain rule of double derivative: Let us consider a two variable function z=f(u,v) where u=p(x,y) and…

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

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Q: Let F = –9zi+ (xe"² – 2xe³) }+ 12 . Find ſg F ·dÃ, and let S be the portion of the plane 2x + 3z = 6…

A: The vector field is F→=-9zi→+xexz-2xex2j→+12k→ The equation of the plane is 2x+3z=6⇒ gx, y=6-2x3.…

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

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Q: Use Stokes's Theorem to evaluate F(x, y, z) = z²i+yj + zk S: z = √4x² - y2 F. dr. In this case, C is…

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Q: Sketch the trace curves of x^2 − y^2 + z^2 − 4x + 2z = 4, and use these trace curves to sketch the…

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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: Let C = C₁ U C₂ U C3, where C₁ is the line segment from (-1, -1) to (0, 0), C₂ is the line segment…

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Q: Show that u(x, y) is harmonic in some domain and find a harmonic conjugate v(x, y) when (a) u(x, y)…

A: Note: As per our guidelines we are supposed to answer only one question. Kindly repost other…

Q: Evaluate € fff.₂²². e² dV where E is enclosed by the paraboloid z = 3 + z² + y², the cylinder z² +…

A: We need to find integral.

Q: In R, let S be the piece of the plane 3x + 3y + 3z = 3 that lies in the first octant, oriented…

A: Given S is the piece of the plane 3x+3y+3z=3 that lies in the first octant oriented upward. The…

Q: Is it true or false that by using a suitable translation x = x' + xo, Y = y' + Yo, it is possible to…

A: Transforming the conic equation

Q: find the flux of f=zk across the portion of the sphere x2+y2+z2=a2 in the first octant in the…

A: Given the field, F=zk The surface S is the portion of the sphere x2+y2+z2=a2 in the first octant in…

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: find the flux of 3 – e")i+ (y³ + sin z)j+ (z³ – unded above by z = V4 – x² outward orientation.

A: We have to find the flux using the divergence therom according to the given information

Q: Find the flux of F = z²k upward through the part of the sphere x² + y² + z² = a² in the first octant…

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Q: Suppose F(x, y) = (2 + 5y, 6z – 5y²). Use Green's Theorem to calculate the circulation of F around…

A: Given : vector field F→x, y=x2+5y, 6x-5y2 and triangle C oriented counter-clockwise with vertices…

Q: find ⌠⌠F*dS if F(x,y,z)=2xi+2yj+2zk and S is the paraboloid z=8-x^2-y^2 lying above the rectangle…

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Q: Use U₁+1,j + Ujj+1 + Uj-1₁j + Ujj-1-4ui, j = 0 to approximate the solution of Laplace's equation at…

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Q: Let r(t)=⟨sin(t),cos(t),9sin(t)+3cos(2t)⟩. Find the projection of r(t)r(t) onto the xz-plane for…

A: Set y=0 as the answer is asked for xz plane.

Q: If F = (2x + y')i + (3y - 4x)j evaluate e F.dr around a triangle ABC in the xy-plane with A(0, 0),…

A: Given  F→=2x+y2i+3y-4xj  we have to evaluate ∮F→.dr→ around  a triangle ABC in the xy-plane with…

Q: Express the given parametric equations of a line using bracket notation and also using i. j.k…

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Q: Verify Green's Theorem in the plane for F = (x² + y²)î + (x² – y²)f in the anti-clockwise direction…

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