Prerequisites for Studying a Book on PDE's
I'm taking a course on Electromagnetism that will cover the boundary-value problems -- the solutions to Laplace's and Poisson's equations for various symmetries, ranging from cartesian to spherically...
View ArticleCasual yet mildly serious Math books to read during reading slump
In the past few months I have been suffering from some problems in my personal life which are now finally over, but I can't get back to doing mathematics like earlier. Please recommend some inspiring...
View ArticleWhat's between the finite and the infinite?
I'm wondering if there are any non-standard theories (built upon ZFC with some axioms weakened or replaced) that make formal sense of hypothetical set-like objects whose "cardinality" is "in between"...
View ArticleDo we need the algebraic structure of $\Bbb R$ to do (algebraic) topology?
Motivation: The Borsuk-Ulam theorem is stated in terms of the antipodal map $x\mapsto-x$, but self-homeomorphisms of the sphere can turn many different maps into $-$, so the theorem should hold if we...
View Articlewarping functions obey diff eq. does that imply $g_t$ obeys same diff eq?
Consider $(M,g_{t})$ equipped with a $1$-parameter family of warped metrics for real parameter $t>0$$$g_{t} = \frac{1}{\phi_t(u)^{2}}\ du^{2} + \phi_t(u)\ dv^{2}$$and suppose that the warping...
View ArticleIntuitive visualisation of a complex vector space
Geometrically, what is a complex vector space?By definition I know it's a vector space which has scalar multiplication defined over the field C complex numbers. But what does this look like...
View ArticleReading slump book recommendation
In the past few months I have been suffering from some problems in my personal life which are now finally over, but I can't get back to doing mathematics like earlier. Please recommend some inspiring...
View ArticleGood Book On Combinatorics
What is your recommendation for an in-depth introductory combinatoric book? A book that doesn't just tell you about the multiplication principle, but rather shows the whole logic behind the questions...
View ArticleIntuition for determinant over other fields/rings [duplicate]
Over $\mathbb R^n$, the standard intuition given to the determinant is that it measures the signed area of the image of an unit cube. But determinants can be more generally defined for endomorphisms of...
View ArticleHow can Line Integrals over a vector field be visualized mathematically?
I've been studying Line Integrals over vector fields; and most books explain that "The Line Integral over Vector Fields gives us the resulting Work done by a vector field over a body/particle moving on...
View ArticleDeterminants after Linear Algebra Done Right
I realize that Axler's Linear Algebra Done Right is a great textbook, and the removal of determinants is often pedagogically sound. However, I was doing some problems on a practice entrance exam, and I...
View ArticlePhrasing an instruction involving the "for all" quantifier
I did a number theory course where the instructor sometimes phrased an exercise in a way that seems slightly wrong. For example, he wrote the exerciseCheck if the following statement is correct: for...
View ArticleDeterminant Intuition in Abstract Vector Spaces over Arbitrary Fields
I was wondering if there was an intuitive way to think about determinants of matrices that represent linear transformations in abstract vector spaces over arbitrary fields.There are many posts about...
View ArticleGood book for self study of a First Course in Real Analysis
Does anyone have a recommendation for a book to use for the self study of real analysis? Several years ago when I completed about half a semester of Real Analysis I, the instructor used "Introduction...
View ArticleConnections between measures and multiplicative functions
Let us define an interesting object as a triple $(P, M, f)$ s.t.$(P, \preceq , ×)$ is a poset equipped with commutative monoid structure s.t. its neutral element $1\in P$ is minimal (i.e. for any $x\in...
View ArticleWhat is going on with the discriminant of $f^{\circ n}(X)-X$?
Recently, I found this amazing answer of @Mercio where they point out the following about the discriminant of the polynomial $P(X)=\frac{f^{\circ 3}(X)-X}{f(X)-X}$ where $f(X)=X^2+c$.We can compute the...
View ArticleCategorical idea of $\mathbb{R}[X]$-Modules
I am a graduate student: I know the basics of Category Theory and Homological Algebra, but nothing too deep.I have spent quite some time studying modules over the ring of real polynomials in one...
View ArticleProof of Weierstrass approximation theorem
I would like to know the proof of the Weierstrass approximation theorem.I prefer proof that the story is easy to follow and requires little prior knowledge, but that is somewhat subjective, so I would...
View ArticleNotation: why $f_\lambda$ in the hook-length formula?
I just learned the hook-length formula: the number of Standard Young Tableaux of size $n$ and shape $\lambda$ is$$\frac{n!}{\prod_\limits{x} \operatorname{hook}(x)}$$where the product runs over all...
View ArticleWhy we need to know how to solve a quadratic?
Five years ago I was tutoring orphans in a local hospital. One of them asked me the following question when I tried to ask him to solve a quadratic:Why do I need how to solve a quadratic? I am not...
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