grandes-ecoles 2015 QII.C.2

grandes-ecoles · France · centrale-maths2__psi Implicit equations and differentiation Gradient computation for multivariable implicit/explicit functions
Deduce that, for all $t \in \mathbb{R}$, the application $(x,y) \mapsto \dfrac{1 - \left((x + \cos t)^2 + (y + \sin t)^2\right)}{x^2 + y^2}$ is harmonic on $\mathbb{R}^2 \backslash \{(0,0)\}$. 
Deduce that, for all $t \in \mathbb{R}$, the application $(x,y) \mapsto \dfrac{1 - \left((x + \cos t)^2 + (y + \sin t)^2\right)}{x^2 + y^2}$ is harmonic on $\mathbb{R}^2 \backslash \{(0,0)\}$.
Paper Questions
QI.A.1 QI.A.2 QI.B.1 QI.B.2 QI.B.3 QI.C.1 QI.C.2 QII.A QII.B.1 QII.B.2 QII.B.3 QII.C.1 QII.C.2 QII.D.1 QII.D.2 QII.D.3 QII.D.4 QIII.A.1 QIII.A.2 QIII.A.3 QIII.B.1 QIII.B.2 QIII.B.3 QIII.C QIV.A.1 QIV.A.2 QIV.A.3 QIV.A.4 QIV.B.1 QIV.B.2 QIV.B.3 QIV.C.1 QIV.C.2