( a) The "vector area" of surface S is defined by a Ξ Jsda. Show that...
a) A vector field F is called incompressible if div F = 0. Show
that a vector field of the form F = <f(y,z),g(x,z),h(x,y)> is
incompressible.
b) Suppose that S is a closed surface (a boundary of a solid in
three dimensional space) and that F is an incompressible vector
field. Show that the flux of F through S is 0.
c)Show that if f and g are defined on R3 and C is a closed curve
in R3 then...
Part B
(4 pts) Consider the integral called the vector area of the surface S. a) Show that ã = 7 for any closed urface. Hint: let (r) = f(F) in the dive gence theorem, where č is a y constant vector. b) Show that (G-F) 4 = 4 x ở Jas for any constant vector c. Hint: let Ā() = (2:) in Stokes' theorem, where is an arbitrary constant vector.
Surface area. Consider a Surface written in the vector form where u and v are parameters (a) Justify or motivate the surface-area formula du dv 1. CiU . (b) Show that the above surface-area formula can also be written as 1.64 where E. F, and G are the coefficients of the first fundamental form. c) Write the surface 1.65 in vector form and show that the above formulas for area imply that S-2
Surface area. Consider a Surface written in...
1. Gauß theorem / Divergence Theorenm Given the surface S(V) with surface normal vector i of the volume V. Then, we define the surface integral fs(v) F , df = fs(v) F-ndf over a vector field F. S(V) a) Evaluate the surface integral for the vector field ()ze, - yez+yz es over a cube bounded by x = 0,x = 1, y = 0, y = 1, z 0, z = 1 . Then use Gauß theorem and verify it....
Figure 2 shows a wedge-shaped closed surface, defined by x + z = 1 and the planes x 0, y 0, y 1, and z 0. For the vector field 1 V xi+yzi+ zk, find the overall flux out of the wedge's surface using Gauss theorem. a) b) According to whether your answer to (a) was positive, negative, or K zero, provide an interpretation of the result.
Figure 2 shows a wedge-shaped closed surface, defined by x + z =...
1-25 By applying the divergence theorem to the special case in which A is a constant but other- wise arbitrary vector, show that the total vector area of a closed surface is zero, that is, $da = 0. Similarly, show that $ds = 0. Do these results surprise you? 1-26 Verify (1-122). (1-123), and (1-124 by using
4. Consider the surface (cone) S given by (a) Calculate the surface area of S (b) Equipping S with an upwards pointing unit normal (one where the z-component of the normal vwctor is positive), calculate the flux of the vector field Fla,,) (x, y, 0) through S , y, z)
4. Consider the surface (cone) S given by (a) Calculate the surface area of S (b) Equipping S with an upwards pointing unit normal (one where the z-component of the...
Evaluate the surface integral ∫∫S F·ds 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. F(x, y, z) = xyl + yzj + zxk S is the part of the paraboloid z = 2 = x2 - y2 that lies above the square 0 ≤ x ≤ 1, 0 ≤ y ≤ 1, and has upward orientation_______
a) What does the solenoidal vector field and irrotational vector field mean, what does it mean physically? Show that in a single mathematical expression, a vector field A is solenoidal and irrotational, respectively. b) A solenoidal field vector along the surface integral of a closed surface is equal to 0 to show through the divergence theorem. c) Show by means of the Stokes theorem that the line integral of an irrotational vector field along the closed curve surrounding a surface...
Calculate the line integral fcs) dr for an arbitrary surface S by using Stoke's theo- rem. Show that for a closed surface S, it holds for every differentiable vector field A(F),
Calculate the line integral fcs) dr for an arbitrary surface S by using Stoke's theo- rem. Show that for a closed surface S, it holds for every differentiable vector field A(F),