Derivation (71 steps)[#96]

Components

(1)Δ(x2+y2+z2)=∇·(∇(x2+y2+z2))Laplacian Scalar Components=∇·((∂∂x(x2+y2+z2)∂∂y(x2+y2+z2)∂∂z(x2+y2+z2)))Gradient Components=∂∂x(∂∂x(x2+y2+z2))+∂∂y(∂∂y(x2+y2+z2))+∂∂z(∂∂z(x2+y2+z2))Divergence Components

Partial

(2)∂∂x(∂∂x(x2+y2+z2))+∂∂y(∂∂y(x2+y2+z2))+∂∂z(∂∂z(x2+y2+z2))=∂∂x(∂∂x(x2+y2)+∂∂x(z2))+∂∂y(∂∂y(x2+y2+z2))+∂∂z(∂∂z(x2+y2+z2))Differentiate Sum=∂∂x(∂∂x(x2)+∂∂x(y2)+∂∂x(z2))+∂∂y(∂∂y(x2+y2+z2))+∂∂z(∂∂z(x2+y2+z2))Differentiate Sum=∂∂x(2x1×∂∂x(x)+∂∂x(y2)+∂∂x(z2))+∂∂y(∂∂y(x2+y2+z2))+∂∂z(∂∂z(x2+y2+z2))Power rule=∂∂x(2x×∂∂x(x)+∂∂x(y2)+∂∂x(z2))+∂∂y(∂∂y(x2+y2+z2))+∂∂z(∂∂z(x2+y2+z2))Power One=∂∂x(2x×1+∂∂x(y2)+∂∂x(z2))+∂∂y(∂∂y(x2+y2+z2))+∂∂z(∂∂z(x2+y2+z2))Differentiate Variable=∂∂x(2x+∂∂x(y2)+∂∂x(z2))+∂∂y(∂∂y(x2+y2+z2))+∂∂z(∂∂z(x2+y2+z2))Multiply One=∂∂x(2x+2y1×∂∂x(y)+∂∂x(z2))+∂∂y(∂∂y(x2+y2+z2))+∂∂z(∂∂z(x2+y2+z2))Power rule=∂∂x(2x+2y×∂∂x(y)+∂∂x(z2))+∂∂y(∂∂y(x2+y2+z2))+∂∂z(∂∂z(x2+y2+z2))Power One=∂∂x(2x+2y×0+∂∂x(z2))+∂∂y(∂∂y(x2+y2+z2))+∂∂z(∂∂z(x2+y2+z2))Differentiate Independent=∂∂x(2x+0+∂∂x(z2))+∂∂y(∂∂y(x2+y2+z2))+∂∂z(∂∂z(x2+y2+z2))Multiply Zero=∂∂x(2x+∂∂x(z2))+∂∂y(∂∂y(x2+y2+z2))+∂∂z(∂∂z(x2+y2+z2))Add Zero=∂∂x(2x+2z1×∂∂x(z))+∂∂y(∂∂y(x2+y2+z2))+∂∂z(∂∂z(x2+y2+z2))Power rule=∂∂x(2x+2z×∂∂x(z))+∂∂y(∂∂y(x2+y2+z2))+∂∂z(∂∂z(x2+y2+z2))Power One=∂∂x(2x+2z×0)+∂∂y(∂∂y(x2+y2+z2))+∂∂z(∂∂z(x2+y2+z2))Differentiate Independent=∂∂x(2x+0)+∂∂y(∂∂y(x2+y2+z2))+∂∂z(∂∂z(x2+y2+z2))Multiply Zero=∂∂x(2x)+∂∂y(∂∂y(x2+y2+z2))+∂∂z(∂∂z(x2+y2+z2))Add Zero=∂∂x(2)×x+2∂∂x(x)+∂∂y(∂∂y(x2+y2+z2))+∂∂z(∂∂z(x2+y2+z2))Product rule=0x+2∂∂x(x)+∂∂y(∂∂y(x2+y2+z2))+∂∂z(∂∂z(x2+y2+z2))Differentiate Constant=0+2∂∂x(x)+∂∂y(∂∂y(x2+y2+z2))+∂∂z(∂∂z(x2+y2+z2))Zero Multiply=0+2×1+∂∂y(∂∂y(x2+y2+z2))+∂∂z(∂∂z(x2+y2+z2))Differentiate Variable=0+2+∂∂y(∂∂y(x2+y2+z2))+∂∂z(∂∂z(x2+y2+z2))Multiply One=2+∂∂y(∂∂y(x2+y2+z2))+∂∂z(∂∂z(x2+y2+z2))Zero Add=2+∂∂y(∂∂y(x2+y2)+∂∂y(z2))+∂∂z(∂∂z(x2+y2+z2))Differentiate Sum=2+∂∂y(∂∂y(x2)+∂∂y(y2)+∂∂y(z2))+∂∂z(∂∂z(x2+y2+z2))Differentiate Sum=2+∂∂y(2x1×∂∂y(x)+∂∂y(y2)+∂∂y(z2))+∂∂z(∂∂z(x2+y2+z2))Power rule=2+∂∂y(2x×∂∂y(x)+∂∂y(y2)+∂∂y(z2))+∂∂z(∂∂z(x2+y2+z2))Power One=2+∂∂y(2x×0+∂∂y(y2)+∂∂y(z2))+∂∂z(∂∂z(x2+y2+z2))Differentiate Independent=2+∂∂y(0+∂∂y(y2)+∂∂y(z2))+∂∂z(∂∂z(x2+y2+z2))Multiply Zero=2+∂∂y(0+2y1×∂∂y(y)+∂∂y(z2))+∂∂z(∂∂z(x2+y2+z2))Power rule=2+∂∂y(0+2y×∂∂y(y)+∂∂y(z2))+∂∂z(∂∂z(x2+y2+z2))Power One=2+∂∂y(0+2y×1+∂∂y(z2))+∂∂z(∂∂z(x2+y2+z2))Differentiate Variable=2+∂∂y(0+2y+∂∂y(z2))+∂∂z(∂∂z(x2+y2+z2))Multiply One=2+∂∂y(2y+∂∂y(z2))+∂∂z(∂∂z(x2+y2+z2))Zero Add=2+∂∂y(2y+2z1×∂∂y(z))+∂∂z(∂∂z(x2+y2+z2))Power rule=2+∂∂y(2y+2z×∂∂y(z))+∂∂z(∂∂z(x2+y2+z2))Power One=2+∂∂y(2y+2z×0)+∂∂z(∂∂z(x2+y2+z2))Differentiate Independent=2+∂∂y(2y+0)+∂∂z(∂∂z(x2+y2+z2))Multiply Zero=2+∂∂y(2y)+∂∂z(∂∂z(x2+y2+z2))Add Zero=2+∂∂y(2)×y+2∂∂y(y)+∂∂z(∂∂z(x2+y2+z2))Product rule=2+0y+2∂∂y(y)+∂∂z(∂∂z(x2+y2+z2))Differentiate Constant=2+0+2∂∂y(y)+∂∂z(∂∂z(x2+y2+z2))Zero Multiply=2+0+2×1+∂∂z(∂∂z(x2+y2+z2))Differentiate Variable=2+0+2+∂∂z(∂∂z(x2+y2+z2))Multiply One=2+2+∂∂z(∂∂z(x2+y2+z2))Zero Add=4+∂∂z(∂∂z(x2+y2+z2))Evaluate Constant=4+∂∂z(∂∂z(x2+y2)+∂∂z(z2))Differentiate Sum=4+∂∂z(∂∂z(x2)+∂∂z(y2)+∂∂z(z2))Differentiate Sum=4+∂∂z(2x1×∂∂z(x)+∂∂z(y2)+∂∂z(z2))Power rule=4+∂∂z(2x×∂∂z(x)+∂∂z(y2)+∂∂z(z2))Power One=4+∂∂z(2x×0+∂∂z(y2)+∂∂z(z2))Differentiate Independent=4+∂∂z(0+∂∂z(y2)+∂∂z(z2))Multiply Zero=4+∂∂z(0+2y1×∂∂z(y)+∂∂z(z2))Power rule=4+∂∂z(0+2y×∂∂z(y)+∂∂z(z2))Power One=4+∂∂z(0+2y×0+∂∂z(z2))Differentiate Independent=4+∂∂z(0+0+∂∂z(z2))Multiply Zero=4+∂∂z(0+∂∂z(z2))Add Zero=4+∂∂z(0+2z1×∂∂z(z))Power rule=4+∂∂z(0+2z×∂∂z(z))Power One=4+∂∂z(0+2z×1)Differentiate Variable=4+∂∂z(0+2z)Multiply One=4+∂∂z(2z)Zero Add=4+∂∂z(2)×z+2∂∂z(z)Product rule=4+0z+2∂∂z(z)Differentiate Constant=4+0+2∂∂z(z)Zero Multiply=4+0+2×1Differentiate Variable=4+0+2Multiply One=4+2Zero Add=6Evaluate Constant

Simplify

(3)6