Let S denote the plane surface whose boundary is the triangle with vertices at (1,0,0), (0, 1,0), and (0,0,1), and let F(x, y, z) = zi+yj + zk. Let n denote the unit normal to S having a nonnegative z-component. Evaluate the surface integral ∬s F.n dS , using (1) The vector representation r(u,v) = (u+v)i + (u-v)j + (1-2u)k. (2) An explicit representation of the form z = f(x,y).

Q: Consider the surface S given by: S : r(u, v) = (uv, u² + v³, 3v²), u = [0, 2], v = [0, ³/] whose…

A:

Q: Let y = Lal: Let L be the line spanned by u. and u %3D Write y as the sum of two orthogonal vectors,…

A:

Q: Let L be the line segment with endpoints at (1,2,3) and (-1,1,4). A light source at is at the point…

A: We will find shadow of given points on the plane given and then find parametric equation of the line…

Q: Your friends correctly calculate the gradient vector for f(x, y)=x²+y at the point (2,-3) as…

A:

Q: H=pz Csup ap te'sin dg tpan 2 At point (1, ,6)

A:

Q: Let F(x, y, z) = xye" – 2° Ja and P(1,0, –1). Let S be the surface defined by the equation xye" –…

A:

Q: Find a unit vector u that is normal to the level curve of f(x, y) = 5x²y through P = (x, y) = (1,-4)…

A:

Q: -y+25 Let F (x, y, z) := x+yz7 3+2⁹ let n be the outward unit normal vector. Evaluate Sf (curl F) ·…

A:

Q: 3. Find a vector function that represents the curve of intersection of the paraboloid z =. and the…

A: To find the vector function that represents the curve of intersection of the paraboloid z=x2+y2, and…

Q: Consider the surface S given by: 4x2 + 4z2 = (1+ y)?, Osys2.Compute: (a) a normal vector at each…

A: Given:-                 Consider the surface S given by: 4x 2 + 4z 2 = (1 + y) 2,0≤y≤2  Compute: (a)…

Q: Let f(x,y, z) = a² + 2y² +z² and g(x,y,z)=z-xy. Find a unit vector u satisfying D.f (1.1,2)= 0 = D,…

A:

Q: F. Sketch the region onto which the sector r ≤ 1,0 ≤ 0 ≤/4 is mapped by the transfor- mation (a) w =…

A:

Q: Check that the point (1,-1,2) lies on the given surface. Then, viewing the surface as a level…

A:

Q: The normal vector to the tangent plane for the surface x²y³ z = 8, at the point (-1, 2, 1) is: 2 3…

A:

Q: Find a vector-valued function, using the given parameter, to represent the intersection of the…

A:

Q: Find a unit normal vector to the surface 6x2 + 2y2 + z2 = 225 at the point P : (5, 5, 5), and…

A:

Q: Let r(u, v)= [2usin(v), 2ucos(v), 2] be a parametrization of a surface S, where 0sus3,0svs2n. Then…

A: Parametrization of a surface

Q: 2. Find a unit normal vector to the surface x + y +2 = 5 at the point (0, 1, 2). Ans. Û + 2k)

A:

Q: y Consider the surface of equation z = X+Y² What is the equation for the tangent line to S at the…

A: This is a problem of vector calculus.

Q: Write formulas for the unit normal and binormal vectors of a smooth space curve r(t)

A: The formulas for unit normal vector  and binomial vectors of a smooth space curve r(t).

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

A:

Q: Vector F is mathematically defined as F = M x N, where M = p 2p² cos + 2p2 sind while N is a vector…

A:

Q: Compute the integral of ſA° · dľ°P2P1 from point P1 (2,1, -1) to P2 (8,2, -1) of the vector function…

A: We have to find the integral from point p1 TP p2.

Q: Let r(u, v)= [2usin(v), 2ucos(v), 2] be a parametrization of a surface S, where 0sus2,0svs27. Then…

A: Given surface is  ,