- #1
yungman
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The definitions of a harmonic function u are: It has continuous 1st and 2nd derivatives and it satisfies [itex]\nabla^2 u = 0[/itex].
Is the second derivative equal zero consider continuous?
Example: [tex] u=x^2+y^2 ,\; \hbox{ 1st derivative }=u_x + u_y = 2x+2y,\; \hbox{ 2nd derivative }=u_{xx} + u_{yy} + u_{xy} + u_{yx} = 2+2=4[/tex] is continuous.
How about [tex] u=x+y,\; \hbox{ 1st derivative }= 1+1=2,\; \hbox{ 2nd derivative }= 0[/tex],
Is the 2nd derivative continuous if equal to zero?
For [itex]\; u=x+y,\; \nabla^2u=0[/itex]!
Is the second derivative equal zero consider continuous?
Example: [tex] u=x^2+y^2 ,\; \hbox{ 1st derivative }=u_x + u_y = 2x+2y,\; \hbox{ 2nd derivative }=u_{xx} + u_{yy} + u_{xy} + u_{yx} = 2+2=4[/tex] is continuous.
How about [tex] u=x+y,\; \hbox{ 1st derivative }= 1+1=2,\; \hbox{ 2nd derivative }= 0[/tex],
Is the 2nd derivative continuous if equal to zero?
For [itex]\; u=x+y,\; \nabla^2u=0[/itex]!
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