GRE Mathematics Subject Test

Flashcards for the GRE Mathematics Subject Test's published content list, split into Calculus, Algebra and Additional topics and then by area, with each topic in a sensible teaching order. They cover definitions, theorems stated with their hypotheses, standard results, counterexamples and short worked results in calculus, linear and abstract algebra, number theory, analysis, topology, complex variables, probability, discrete mathematics and numerical analysis.

Adding it gives you your own copy, with every subdeck below. Each card then comes back just before you would forget it, and every one you get right or wrong counts towards your mastery of its topic. You can delete or suspend the parts you are not studying once it is yours.

Some of its cards

• What conditions define a topology $\tau$ on a set $X$?

  $\varnothing,X\in\tau$; arbitrary unions of members lie in $\tau$; finite intersections of members lie in $\tau$
• What four axioms define a group $(G,\ast)$?

  Closure, associativity, an identity element, and an inverse for every element
• Define $f'(a)$ as a limit.

  $f'(a)=\lim_{h\to0}\frac{f(a+h)-f(a)}{h}$
• What is a ring?

  An abelian group under addition with an associative multiplication that distributes over addition on both sides
• What is the integrating factor for $y'+P(x)y=Q(x)$?

  $\mu(x)=e^{\int P(x)\,dx}$
• When does $\sum_{n\ge0}ar^n$ converge, and to what?

  When $|r|<1$, to $\frac{a}{1-r}$
• Is matrix multiplication commutative?

  No, $AB\ne BA$ in general
• State the epsilon-delta definition of $\lim_{x\to a}f(x)=L$.

  For every $\varepsilon>0$ there is a $\delta>0$ such that $0<|x-a|<\delta$ implies $|f(x)-L|<\varepsilon$
• State the rank-nullity theorem for a linear map $T:V\to W$ with $V$ finite-dimensional.

  $\dim V=\dim\ker T+\dim\operatorname{im}T$
• What is the degree $[K:F]$ of a field extension?

  The dimension of $K$ as a vector space over $F$

And 832 more once you add the deck.