6. (6 points) Let u = u(,y) be a smooth function in R2 such that au...
Problem 3 (12 points): Let D be a bounded domain in R" with smooth boundary. Suppose that K(x, y) is a Green's function for the Neumann . For each x E D, the function y H K(x, y) is a smooth harmonic For each x E D, the normal derivative of the function y K(x, y) . For each z e D, the function y K(x,y)-Г(z-y) is smooth near problem. This means the following: function on D(r satisfies (VyK(x, y).v(b))-arefor...
Implicit Function Theorem in Two Variables: Let g: R2 → R be a smooth function. Set {(z, y) E R2 | g(z, y) = 0} S Suppose g(a, b)-0 so that (a, b) E S and dg(a, b)メO. Then there exists an open neighborhood of (a, b) say V such that SnV is the image of a smooth parameterized curve. (1) Verify the implicit function theorem using the two examples above. 2) Since dg(a,b) 0, argue that it suffices to...
please be as detailed as possible
Question 5, Let ơ (u, v) : R2- R3 be a smooth function (not necessarily a surface patch). Let E Ou .Ou, F-Ou . συ and G Oy .Oy. Show that the following equalities hold: (Here D denotes total derivative.)
Question 5, Let ơ (u, v) : R2- R3 be a smooth function (not necessarily a surface patch). Let E Ou .Ou, F-Ou . συ and G Oy .Oy. Show that the following equalities...
2. Let U C R2 be simply connected and let to E U. Let g: U(oR2 be irrotational and of class C1. Assume that there exists r >0 such that B(zo, r) C U and g=0. Let γ be a closed sinile polygonal arc with range in U \ {zo), let「be its range, and let V be the bounded connected component of R2 \ Г. (a) Assume that V C U \ [xo) and prove that g=0. (b) Assume that...
(5 marks) Consider a smooth function u(x, y) satisfying: Show that u attains its maximum on the boundary àS2
(5 marks) Consider a smooth function u(x, y) satisfying: Show that u attains its maximum on the boundary àS2
answer all parts, please!
(5) Consider the closed volume V contained by the cylinder r2+2-4 and the planes y =-2 and r +y-3. Let the surface S be the boundary of this region. Note that this boundary consists of three smooth pieces. (a) Clearly sketch and label S. (You may use GeoGebra for this.) (b) In complete sentences, verbally describe what this surface looks like. (c) Find a parametric representation for each of the three parts of the boundary S...
Calculus
. Let h(x, y) be a smooth parametrization on a region H for a surface S in R3. Suppose there is a continuous transform F :R + H, (u, v) + x(u, v), y(u, v)) such that F is one-to-one on the interior of the region R and r;=ho F is a smooth parametrization on R for S. Show that 9 S/ \ru xroldA= S/ \he x hy|dA= A(s where A(S) is the area of S. (15 pts] 9
Can anyone help with this question please? The initial boundary
condition is trivial, I struggled to show the first condition.
Any help will be appreciated!!!
Let now Ω c Rd be an open and bounded set with a smooth boundary on and outer unit normal n. Furthermore, let f : Ω → R be a continuous function. Define the functional where weg-(u E C10) |v = 0 and ▽u . n = 0 in 201. Show that a minimiser u...
Problem 1: Let F(, y,) be a function given by F(, y, z) (r2+y)e. Let S be the surface in R given by the equation Fr, y, 2) 2. (a) Find an equation of the tangent plane to the surface S at the point p(-1,1,0) (b)Find the directional derivative -1,1,0) of F(,y,2) in the direction of the unit vector u = (ui, t», t's) at the point p(-1,1,0) - In what direction is this derivative maximal? In what direction is...
Problem 2. Let be the quarter torus with outward normal. Use the parameterization r(u, v) = (4 + 2 cos(v)) cos(u)i + (4 + 2 cos(u)) sin(u)j + 2 sin(v)k, for 0 Susand 0 <0527 (a) Find a parameterization for each of the curves forming the boundary of E. Make sure the orientation of the curves match the orientation induced by S. (b) Let F(x, y, z) = xyi+yzj+rzk. Evaluate S/.( VF) ds.