Advanced Calculus fi..

Chapter 2 Differential Calculus of Functions of Several Variables1 .)c) 3 and & if w = eX--y-,~ I I I + # I ~ , ,d) 1f w = xmyn.Formula (2.138) can be used as a check for (a) and (b).2. Verify that the mixed derivatives are identical for the following cases:a22, Za) w. and & forz = *,x- + ya7w a3w aJu,b) a n , m. and for w = Jx2 + y2 + z2.3. Show that the following functions are harmonic in x and y:a) ex cos y, b) x3 - 3xy2, C) log Jm.4. a) Show that every harmonic function is biharmonic.b) Show that the following functions are biharmonic in x and y:xeX cosy, x4 - 3x2y2.c) Choose the constants a, b, c so that ax2 + bxy + cy2 is harmonic.d) Choose the constants a, b, c, d so that ax3 + bx2y + cxy2 + dy3 is harmonic.5. a) Prove the identity:v2(uv) = uv2v + vv2u + 2vu . v vfor functions u and v of x and y.b) Prove the identity of (a) for functions of x, y, and z.c) Prove that if u and v are harmonic in two or three dimensions, thenis biharmonic. [Hint: Use the identity of (a) and (b).]d) Prove that if u and v are harmonic in two or three dimensions, thenis biharmonic, where r2 = x2 + y2 for two dimensions and r2 = x2 + y2 + z2 forthree dimensions.6. Establish a chain rule analogous to (2.133) for a2z/au av.7. Use the rule (2.133), applied to a2w/ax2 and a2w/ay2, to prove (2.138).8. Prove (2.140). [Hint: Use (2.139) to express v2w in cylindrical coordinates; then notethat the equations of transformation from (z, r) to (p, 4) are the same as those from (x, y)to (r, 01.19. Prove that the biharmonic equation in x and y becomesin polar coordinates (r, 8). [Hint: Use (2.138).]10. If u and v are functions of x and y defined by the equationsfind a2u/ax2.xy+uv=l,xu+yv=l,