A) Find the surface area of the paraboloid when by parameterizing the surface using a polar representation.
B) Express the surface area of the paraboloid as a triple integral using the Divergence theorem.
C) By choosing an appropriate vector field, use the Divergence Theorem to find the surface area of the paraboloid.
A) Find the surface area of the paraboloid when by parameterizing the surface using a polar representation. B) Expres...
(1) Let P denote the solid bounded by the surface of the hemisphere z -Vl-r-y? and the cone2y2 and let n denote an outwardly directed unit normal vector. Define the vector field F(x, y, z) = yi + zVJ + 21k. (a) Evaluate the surface integral F n dS directly without using Gauss' Divergence Theorem. (b) Evaluate the triple integral Ш div(F) dV directly without using Gauss' Diver- gence Theorem Note: You should obtain the same answer in (a) and...
A) Find a vector valued function of the form for the paraboloid . B) Find a vector valued function for the elliptic cylinder r(u, v) = x(u, v)i + y(u, v)j + z(u, v)k We were unable to transcribe this imageWe were unable to transcribe this image
(7) Let V be the region in R3 enclosed by the surfaces+2 20 and z1. Let S denote the closed surface of V and let n denote the outward unit normal. Calculate the flux of the vector field F(x, y, z) = yi + (r2-zjy + ~2k out of V and verify Gauss Divergence Theorem holds for this case. That is, calculate the flux directly as a surface integral and show it gives the same answer as the triple integral...
Let R be delimited by and and S being surface R, outwardly. Now give us the vector field F(x,y,z)=ij + calculate flux integral We were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this image(z + sin ( 2)) +(y + cos(r3 +(22 + sin(zy))k
a) What is the Surface Integral b) What is the Triple Integral Verify the Divergence Theorem for the vector field F(x, y, z) = (y,1,22) on the region E bounded by the planes y + 2 = 2, 2= 0 and the cylinder r2 + y2 = 1.
Question 3. Let Q be the solid hemisphere bounded by x + y² + 2 = 1 for 2 > 0 and by the plane z = 0, and let F = xi+yi + zk be a vector field. Verify the divergence theorem for Q and F by answering parts (a) and (b) below. Part (a) (5 points). Find the value of the triple integral of the divergence of F over the solid hemisphere Q. Part (b) (10 points). Evaluate...
B.2. The surface Sc of an ice-cream cone can be parametrised in spherical polar coordinates (r, 0, 0) by where θ0 is a constant (which you may assume is less than π/2) (a) Sketch the surface Sc (b) Using the expression show that the vector element of area on Sc is given by -T Sin where [41 (c) The vector field a(r) is given in Cartesian coordinates by Show that on Sc and hence that 4 2 (d) The curved...
10. Use Gauss Divergence Theorem to find the flux for a flow field with v-(r')i+(y3)/t(e)k through the surface of a solid constructed by slicing the cylinder + y 9 with the plane x+z-5.Clearly construct the triple integral of the order dz dy dx but you do not need to evaluate it x+z-5 10. Use Gauss Divergence Theorem to find the flux for a flow field with v-(r')i+(y3)/t(e)k through the surface of a solid constructed by slicing the cylinder + y...
Find the area (surface area) of the part of the hyperbolic paraboloid z = y2 - x that lies between the cylinders x + y2 = 1 and x² + y2 = 4
5. Let E be the solid bounded by the paraboloid y = x2 + z2 , the cylinder x2 + z2 = 1, and the plane y = 2. Let S be the surface of E with outward orientation. (b) Evaluate the volume integral FX,Y,Z) = yj + zk We were unable to transcribe this image