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- Find an orthonormal basis for the subspace of Euclidean 3 space below. W={(x1,x2,x3):x1+x2+x3=0}se the Divergence Theorem to evaluate S F · N dS and find the outward flux of F through the surface of the solid bounded by the graphs of the equations. Use a computer algebra system to verify your results. F(x, y, z) = x3i + x2yj + x2eyk S: z = 4 − y, z = 0, x = 0, x = 3, y = 0Double integrate under z=xy, above the triangle with vertices (0,1),(0,4),(1,1).
- F is the three-dimensional vector field defined by F(x,y,z) = ( z, x, y ). In other terms, P(x,y,z) = z, Q(x,y,z) = x, R( x,y,z) = y. Also, the domain T is the equilateral triangle in space with vertices (corners) at ( 2, 0, 0 ), ( 0, 2,0 ), ( 0, 0, 2 ), and normal vector n = ( a, b, c ) with a > 0 AND b > 0 AND c > 0. The boundary, or perimeter, C of the triangle T consists of three straight segments, oriented counterclockwise from (2,0,0) to (0,2,0), then from (0,2,0) to (0,0,2), and back from (0,0,2) to (2,0,0). Question: Calculate the circulation (line integral) of F around the boundary C of the triangle T.Fs<x, y, z> = < 0, y, -z >, S consists of the paraboloid y = x2 + z2 , 0 < y < 1, and the disk x2 + z2 < 1, y = 1. Evaluate the surface integral for the given vector field F and the oriented surface S. In other words, find the flux of F across S. For closed surfaces, use the positive (outward) orientation.Divergence Theorem for more general regions Use the DivergenceTheorem to compute the net outward flux of the following vectorfields across the boundary of the given regions D.
- The flux of the vector field F(x,y,z)=3zk across the closed rectangular box with opposing vertices at (0,0,0) and (6,5,7) is: Using surface integrals, not divergence theorem. Note: Next week we learn a theorem that will make this problem much easier, but for now you need to work out a surface integral and show your work or explain your geometric reasoning.F is the three-dimensional vector field defined by F(x,y,z) = (x,y,z). In other terms, P(x,y,z) = x, Q(x,y,z) = y, R( x,y,z) = z. Also, the domain D = [0,1]x[0,1]x[0,1] is the solid unit cube, which consists of every (x,y,z) such that 0 <= x <= 1 AND 0 <= y <= 1 AND 0<= z <= 1. The surface S of the cube D consists of six square faces, with normals pointing out of the cube. Question: Calculate the flux of F across the surface S of the cube D.Let Q be the solid region bounded by the surface S of equation given by z2 = x2 + y2 , 0 ≤ z ≤ 3, and the cylinder of equation x2 + y2 = 9,0 ≤ z ≤ 3.Find the outward flux of the vector field G(x, y, z) = x2 i+ y2 j+ z2 k through the surface S where S is the surface of Q.
- If ϕ=3xy+2x2yz−12xz3ϕ=3xy+2x2yz−12xz3, Find ∇×(∇ϕ)The vector field F(x,y)=xi+yjx2+y2F(x,y)=xi+yjx2+y2 is solenoidal what points? Find the divergence at (4,0,0)(4,0,0) for the vector field F(x,y,z)=exsinyi−excosyj+z2kF(x,y,z)=exsinyi−excosyj+z2k Consider the conservative vector field F(x,y,z)=(yz+2x)i+(xz+2y)j+(xy+2z)k.F(x,y,z)=(yz+2x)i+(xz+2y)j+(xy+2z)k. Evaluate ∫cF⋅dr,∫cF⋅dr, c: is a curve starting at (0,0,0)(0,0,0) to (1,1,1).(1,1,1). Use Stoke's theorem to evaluate ∮c(sinzdx−cosxdy+sinydz)∮c(sinzdx−cosxdy+sinydz) where c is the boundary of the rectangule 0≤x≤π,0≤y≤1,z=3. Determine whether the vector field is conservative. If it is, find a potential function for the vector field. F(x,y,z)=7x6y8z9i+8x7y7z9j+9x7y8z8kF(x,y,z)=7x6y8z9i+8x7y7z9j+9x7y8z8k Given a vector field F(x,y,z)=(3x+z)i+(y2−sinx2z)j+(xz+yex5)k,F(x,y,z)=(3x+z)i+(y2−sinx2z)j+(xz+yex5)k, the divengence theorem value in the region…(a)Show that F is a conservative vector field. f(x,y,z) = < yzcos(xz)-y^(2), sin(xz) - 2xy, 3z^2 + xycos(xz) > (b)Find a potential function for F.Help ASAP A. From the given vector field F⃗ (x, y, z) = < 2z3 − 4y2 + axy , x2 − 8xy − bz , 6xz2 − ay + 4 >, compute the curl of F⃗ and find the values of a and b that will turn F⃗ into a conservative vector field. B. Find the area of the portion of the cylindrical surface y = 4 − x2 in the first octant that is between the xy-plane and the surface x2 + 2x + y − z = 4.