Advanced book on Mathematics Olympiad

3.3 Multivariable Differential and Integral Calculus 167 ∫ 1 0 e^2 πimxdx= 1 2 πim e^2 πimx ∣ ∣∣ ∣ 1 0 = 0. Therefore, equality ...

168 3 Real Analysis means that its graph (which is a surface inR^3 ) admits a tangent plane at each point. For a three-variable ...

3.3 Multivariable Differential and Integral Calculus 169 for somek≤n. Multiply this equality byn, then apply the operatorx∂x∂ +y ...

170 3 Real Analysis If a differentiable multivariable function has a global extremum, then this extremum is found either among t ...

3.3 Multivariable Differential and Integral Calculus 171 f (x, y)=x^4 + 6 x^2 y^2 +y^4 − 9 4 x− 7 4 y. 503.Find the equation of ...

172 3 Real Analysis off a horizontal mirror represented schematically in Figure 22. Denote byCandD the projections ofAandBonto t ...

3.3 Multivariable Differential and Integral Calculus 173 Solution.We will show that the best choice forkis^14. To prove this fac ...

174 3 Real Analysis 508.Using the method of Lagrange multipliers prove Snell’s law of optics: If a light ray passes between two ...

3.3 Multivariable Differential and Integral Calculus 175 There are three special situations worth mentioning: The change in two ...

176 3 Real Analysis V= 1 2 ∫∞ 0 t^2 t^4 + 1 dt. A routine but lengthy computation using Jacobi’s method of partial fraction deco ...

3.3 Multivariable Differential and Integral Calculus 177 518.Prove the Gaussian integral formula ∫∞ −∞ e−x 2 dx= √ π. 519.Evalua ...

178 3 Real Analysis ∫b a ∑∞ n= 0 f (n, x)= ∑∞ n= 0 ∫b a f (n, x). Here we are allowed to commute the sum and the integral if eit ...

3.3 Multivariable Differential and Integral Calculus 179 523.Show that fora, b >0, ∫∞ 0 e−ax−e−bx x dx=ln b a 524.Let|x|<1 ...

180 3 Real Analysis The Gauss–Ostrogradsky (Divergence) Theorem.LetSbe a smooth, orientable sur- face that encloses a solid regi ...

3.3 Multivariable Differential and Integral Calculus 181 ∮ C ydx+zdy+xdz=−A √ 3 , whereAis the area of the disk bounded byC. Obs ...

182 3 Real Analysis As a corollary, we obtain the well-known fact that a container filled with gas under pressure is at equilibr ...

3.3 Multivariable Differential and Integral Calculus 183 We apply this to F(z)= f(z) z−a = (u(x, y)+iv(x, y))(x−iy+α−iβ) (x+α)^2 ...

184 3 Real Analysis 530.Letf, g:R^3 →Rbe twice continuously differentiable functions that are constant along the lines that pass ...

3.4 Equations with Functions as Unknowns 185 534.For two disjoint oriented curvesC 1 andC 2 in three-dimensional space, parametr ...

186 3 Real Analysis But we like more the nonstandard functional equations. Here is one, which is a simplified version of a short ...