(b) Show that -(y-and toTR anid with where r = 1Ti (b) Show that -(y-and toTR anid with where r = 1Ti
If z = f(x, y), where x = r + s and y=r – s, show that 2 2 дz Әх дz ду дz дz მr as
Let h(x, y) = In r where r = V x2 + y2. Show that Ꭷh , ch . Ꭷ2 + Ꭷ2 = 0.
3. If z = f(x,y), where x = r cos, y=r sin 0 show that 222 222 1 222 1.az + + +) ar2 ду? ar2 a02 rar
1. (a) Prove that R,R C R,, and then (b) use it to show that y in R and E R,then l lel:] and (c) as a consequence, r]1> when 0< r [y).
11. (8 marks) Let F(x, y, z) = x'yz, where r, y,z E R and y, z 2 0. Execute the following steps to prove that F(z,y,2) < (y 11(a) Assume each of r, y, z is non-zero and so ryz= s, where s> 0. Prove that 3 F(e.y.) (y)( su, y su, z sw and refer back to Question (Hint: Set 10.) 11(b) Show that if r 0 or y0 or z 0, then F(z, y, z) ( 11(c)...
7. Consider the differential equation (a) Show that z 0 is a regular singular point of the above differential equation (b) Let y(x) be a solution of the differential equation, where r R and the series converges for any E (-8,s), s > 0 Substitute the series solution y in to the differential equation and simplify the terms to obtain an expression of the form 1-1 where f(r) is a polynomial of degree 2. (c) Determine the values of r....
you can skip #2
Show that F() = Vf (), 1. Let F R3 -R be defined by F(I) = F12", where u where f(r,y,) = =- +22 2. Consider the vector field F(E,) = (a,y) Compute the flow lines for this vector field. 3. Compute the divergence and curl of the following vector field: F(x,y,)(+ yz, ryz, ry + 2)
Show that F() = Vf (), 1. Let F R3 -R be defined by F(I) = F12", where u...
Assume u E C2 (B (0,r)) solves the boundary-value problem where g E C(OB(0,r)). Show that gry.ndS(y) (хев"(0. т.)) which is called Poisson's formula with Poisson's kernel
Assume u E C2 (B (0,r)) solves the boundary-value problem where g E C(OB(0,r)). Show that gry.ndS(y) (хев"(0. т.)) which is called Poisson's formula with Poisson's kernel
3. Evaluate each of the following for the universe 2 (a) 3rvy <y where r,yEZ (b) Vyar <y where r,y E (d) Vrvy yty where a,VEZ
(P(x),Q(y), R(z)), where P depends only 2. Let S be any surface with boundary curve C, and let F(x,y, z) on r, where Q depends only on y, and where R depends only on z. Show that F.dr 0 C
(P(x),Q(y), R(z)), where P depends only 2. Let S be any surface with boundary curve C, and let F(x,y, z) on r, where Q depends only on y, and where R depends only on z. Show that F.dr 0 C