/sci/ - Science & Math

>>15026612
Your soul is a number whose digits encode your brain's internal computation within some computing cellular automaton, say Conway's game of life or automaton 110. It's an object in plato's realm, you are that number.

There is a breeding property of mental computation that can be transferred to let your soul infect others, and due to your abstract-ness you will continue in the realm of mathematics even when your physical body dies, or is insufficiently competent to see the whole picture.

Is this kind of what open individualism is talking about? I don't really get it.

>>12027119
relate the derivatives to each other with chain rule and it all falls through

[eqn]
\frac{\partial}{\partial x_i}
= \sum_j \frac{\partial x'_j}{\partial x_i} \frac{\partial}{\partial x'_j}
[/eqn]
this relates the definition of the gradient in two (completely arbitrary) bases. writing this matrix notation we get
[eqn]
\nabla = \mathbf{J} \nabla'
[/eqn]
where [math]\mathbf{J}[/math] is the jacobian (i.e. the coordinate transformation).

next apply the definition of divergence to get laplacian
[eqn]
\nabla^2 = \nabla \cdot \nabla = (\mathbf{J} \nabla') \cdot \mathbf{J} \nabla' = \nabla^T \mathbf{J}^T \mathbf{J} \nabla'
[/eqn]
assuming an orthogonal coordinate transformation matrix [math]\mathbf{J} \in \{\mathbf{O} \in \mathbb{R}^{n \times n} | \mathbf{O}^T \mathbf{O} = \mathbf{I}\}[/math], then the above equation simplifies
[eqn]
\nabla^2 = \nabla'^T \mathbf{I} \nabla' = \nabla'^T \nabla' = \nabla' \cdot \nabla' = \nabla'^2
[/eqn]
thus
[eqn]
\nabla^2 = \nabla'^2
[/eqn]

be careful when manipulating derivative operators. you really should give it a function to work on so you don't miss any applications of the product rule, etc.