Dimension of spaces

1. Plot R: (filled triangle)
2. Plot $\partial R$: (empty triangle like $\nabla$)
3. @param $\partial R$
   1. We have 3 lines that we @param: $r_1(t)=t<1,1>+<0,0>r_2(t)=t<1,0>+<1,1>r_3(t)=t<1,-1>+<0,0>$
   2. As long as our secondary part is some point on line works
4. What’s the ${\int }_{?}xd?=?$
   1. Transform $x=|r_i^{\prime }(t)|$
   2. ${\int }_{r_i(t)}|r_i^{\prime }(t)|dt+{\int }_{r_{i+1}(t)}|r_{i+1}^{\prime }(t)|dt+$
   3. In this case, remembering to account for the jacobian we have: $${\int }_{r_1(t)=[0,1]}|r_1^{\prime }(t)|dt={\int }_0^1|t\sqrt{2}|dt+\dots$$
5. What’s $\int {\int }_{?}xd?=?$
   1. a

Thinking about object dimensions?

@Param

• $S(s)=<s,s+1>$
• LHS = dim of object
• RHS = dim of space
• So more commas in RHS ++ space
• more vars LHS ++ object

Equastions

• Say we have $x^2+y^2=4$
  • 1D object in 2D space
• Wht about both
  • $x^2+y^2=4x-y=4$ → 0D object in 2D space
• Wht about $x^2+y^2+z^2=1?$ 2D in 3D space!