
Let S CR3 be the regular surface given by z = x - 3ry?. a) Calculate...
Let R^2 be equipped by the metric ds^2 = (4/(1 + x^2 + y^2 )^2)
(dx^2 + dy^2 ), i.e. its first fundamental form is E = G =
4/(1+x^2+y^2)^2 , F = 0. Use the formula dω12 = −Kω1 ∧ ω2 to
calculate its Gauss curvature.
Let R2 be equipped by the metric i.e. its first fundamental form is E = G = TrtFF, F = 0. Use the formula dui,-- . КМ Л w2 to calculate its Gauss...
(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 S be the surface S ((x, y, z) ER3z 7y2 0 (i) Show that the function a :R2-R3, , given by a(t, u)- (t 2,3ut, 7u2), is C1 on all of R2 and satisfies a(t,u) E S for all (t,u) E R2 ii) Show that a is not injective. (ii) Find all the points of the domain where Da is not injective.
4. Consider the equation zy" - 2y' y 0 (a) Explain why r 0 is a regular singular point for the given equation (b) Let ri >r2 be two indical roots of the given equation. Using Frobenius' method, find a series solution n(x)-z"Ση_0Cnz". (c) Find the second solution of the form Σ000 bnXntr2 with boメ0, or i (z) Inr +bn+r2 with the first three nonzero terms of the series with coefficient bn
4. Consider the equation zy" - 2y' y...
Let S be the surface of the box given by {(x, y, z)| – 2 < x < 0, -1 <y < 2, 0 Sz<3} with outward orientation. - Let F =< – xln(yz), yln(yz), –22 > be a vector field in R3. Using the Divergence Theorem, compute the flux of F across S. That is, use the Divergence Theorem to compute SSF. ds S
Let S be the surface of the box given by {(x, y, z) – 2 <<<0, -1<y<2, 0<z<3} with outward orientation. Let Ę =< -æln(yz), yln(yz), –22 > be a vector field in R3. Using the Divergence Theorem, compute the flux of F across S. That is, use the Divergence Theorem to compute SS F. ds S
surface patch for S. regular surface and f: S Ra smooth EXERCISE 3.44. Let S be a function. Assume that the point p e S is a critical point of f, which means that dfp(v) 0 for all v e TpS. Define the Hessian of f atp in the direction v as Hess(f)p(v) (foy)"(0), where y is a regular curve in S with y(0) = p and y'(0) = v. Prove that the Hessian is well defined in the sense...
use divergence theorem
Let S be the surface of the box given by {(x, y, z)| – 1 < x < 2, 05y<3, -2 << < 0} with outward orientation. Let F =< xln(xy), –2y, –zln(xy) > be a vector field in R3. Using the Divergence Theorem, compute the flux of F across S. That is, use the Divergence Theorem to compute SSĒ.ds S
Let F(x,y,z) = 4i – 3j + 5k and S be the surface defined by z= x2 + y2 and 22 + y2 < 4. Evaluate SJ, F. nds, where n is the upward unit normal vector.
te tt 9 Let S be the surface defined by r2 y-22= 1 and 0 15 points by the normal direction toward the z-axis. Find the flux of the velocity field V 1and oriented (z2-ry2)i+(2.2y -yz2)j+ (y2z- 2r2)k across S Solution. To use Gauss Theorem, define C and C2 such that C1 {(a, y, z ) | a + y? < 2, z = 1} C2={(r,y,)| +y1.0} 11 TALK X W P S ww F6 & # 2 3 4...