Geometric topology | 3-manifolds
In low-dimensional topology, the trigenus of a closed 3-manifold is an invariant consisting of an ordered triple . It is obtained by minimizing the genera of three orientable handle bodies — with no intersection between their interiors— which decompose the manifold as far as the Heegaard genus need only two. That is, a decomposition withfor and being the genus of . For orientable spaces, ,where is 's Heegaard genus. For non-orientable spaces the has the form depending on theimage of the first Stiefel–Whitney characteristic class under a Bockstein homomorphism, respectively for It has been proved that the number has a relation with the concept of , that is, an orientable surface which is embedded in , has minimal genus and represents the first Stiefel–Whitney class under the duality map , that is, . If then , and if then . (Wikipedia).
Projection of One Vector onto Another Vector
Link: https://www.geogebra.org/m/wjG2RjjZ
From playlist Trigonometry: Dynamic Interactives!
Trig identities - What are they?
► My Trigonometry course: https://www.kristakingmath.com/trigonometry-course Trig identities are pretty tough for most people, because 1) there are so many of them, and 2) they’re hard to remember, and 3) it’s tough to recognize when you’re supposed to use them! But don’t worry, because
From playlist Trigonometry
From playlist Trigonometry TikToks
From playlist Trigonometry TikToks
Use pythagorean identities to verify an identity
👉 Learn how to verify Pythagoras trigonometric identities. A Pythagoras trigonometric identity is a trigonometric identity of the form sin^2 (x) + cos^2 (x) or any of its derivations. To verify trigonometric expression means to verify that the term(s) on the left-hand side of the equality
From playlist Verify Trigonometric Identities
Adding Vectors Geometrically: Dynamic Illustration
Link: https://www.geogebra.org/m/tsBer5An
From playlist Trigonometry: Dynamic Interactives!
Using pythagorean identities to help me verify an identity
👉 Learn how to verify Pythagoras trigonometric identities. A Pythagoras trigonometric identity is a trigonometric identity of the form sin^2 (x) + cos^2 (x) or any of its derivations. To verify trigonometric expression means to verify that the term(s) on the left-hand side of the equality
From playlist Verify Trigonometric Identities