Top Ten Theorems: Metric Geometry Part 1

The Top Ten Handy and Fun theorems of metric geometry. Part 1: Pythagorean theorem, Symmetric anti-symmetric decomposition theorem, Proof that the metric tensor is symmetric. Einstein notation is introduced and used in some of these proofs.

Category:

Education

Tags:

• Metric
• Geometry
• Theorems
• Tensor
• Analysis
• Einstein
• summation
• convention
• symmetric
• tensor
• metric

[ForgedInASupernova]

Analysis, Diff. Eqs, Metrics, Tensors Matrices,...

i.e.

God's sentences.

:)

[Mathview]

@ForgedInASupernova TY.... At a place I used to work the campus store sold teeshirts with this phrase printed on front and back: And God Said: " divE=rho, divB=0 curlE=-dB/dt curlB = J + dE/dt " ... and there was Light. (I believe they used SI units so there were factors of c, mu0 and eps0 in the right places). Another version had the the same phrase but with the Schrodinger Eq., and E = h*nu, h-bar k = omega for QM fans. We like those shirts.

[skbpolo]

i like this a little bit 2 much. it makes me create bodily

[Mathview]

@skbpolo It's ok to like it too much. ;)

[h1451]

hey 3:49 can't understand why you added "2ab" on the other side? aren't you only simplifying the right side of the equation?? so it should be "a^2+b^2=c^2+2ab" ?? am i wrong? help me i;m confused @.@

[Mathview]

@h1451 I missed your comment some how. So anyway, here's the thing... The sides of the large square have length (a+b) and four triangles have area 4ab/2 so when you add the 4 triangles to the inner square (c^2) that must equal the outer square (a+b)^2. Equating the two areas lets you "cancel-out" the terms 2ab from both sides.

[ForgedInASupernova]

@Mathview Haha a friend of mine has one of those shirts with the former on it. I want one with schrodinger's thats one of my favorites hah.