Intereting Posts

When can't $dy/dx$ be used as a ratio/fraction?
What is the origin of the formula: $\rho_x (x)=\left|\frac{{d}x}{{d}\alpha}\right|^{-1}\rho_\alpha(\alpha)$ that relates random variables?
Is the quotient $X/G$ homeomorphic to $\tilde{X}/G'$?
About closure under +
subgroup of order $11$ lies inside $Z(G).$
$p_n(x)=p_{n-1}(x)+p_{n-1}^{\prime}(x)$, then all the roots of $p_k(x)$ are real
In how many words the letter of word RAINBOW be arranged so that only 2 vowels always remain together?
Physical or geometric meaning of the trace of a matrix
Prove that a planar graph is connected if it has $p$ vertices and $3p-7$ edges
Counting squarefree numbers which have $k$ prime factors?
If $|\det(A+zB)|=1$ for any $z\in \mathbb{C}$ such that $|z|=1$, then $A^n=O_n$.
Iterated Limits Schizophrenia
n-tuple Notation
Uniform Spaces: Completeness
Finding sum $\sum_{i=1}^n \frac1{4i^2-1}$

So I’m wanting a solid math book for Christmas. I have a solid background in Calculus and am currently working through baby Rudin. I really want a rigorous book dealing with multivariable calculus and linear algebra. How well does Apostol II do this? Would this be a good continuation book from my current study? If not, what book would you suggest?

EDIT: I really like Apostol’s Multivariable Calculus and Linear Algebra book and its format. Will this book accomplish my purpose? Do you think it is a good book?

- Proving the last part of Nested interval property implying Axiom of completeness
- How many roots have a complex number with irrational exponent?
- Discretization of an integral
- How can I show using the $\epsilon-\delta$ definition of limit that $u(x,y_0)\to l$ as $x\to x_0?$
- How can I compute the integral $\int_{0}^{\infty} \frac{dt}{1+t^4}$?
- Integrating $e^{a/x^2-x^2}/(1-e^{b/x^2})$

- equation involving the integral of the modular function of a topological group
- Calculating the determinant gives $(a^2+b^2+c^2+d^2)^2$?
- Number of edges of any spanning tree of a graph $G$ coming from a subgraph of $G$
- What are the eigenvalues of matrix that have all elements equal 1?
- How do we show every linear transformation which is not bijective is the difference of bijective linear transforms?
- How to find the exact value of $ \cos(36^\circ) $?
- Find $\lim (a_{n+1}^\alpha-a_n^\alpha)$
- Relationship between eigendecomposition and singular value decomposition
- Prove that T is a linear transformation
- An orthonormal set cannot be a basis in an infinite dimension vector space?

Ted Shifrin’s book Multivariable Mathematics is actually one of the best books of this type while not being very well known. Unfortunately, it’s very expensive, so unless you can find it in your library, I would choose something else.

Otherwise I would just recommend Spivak’s Calculus on Manifolds together with some linear algebra book. For linear algebra I would recommend either Axler’s Linear Algebra Done Right or Linear Algebra by Fiedberg, Insel and Spence

The best book on this for beginners is John and Barbara Hubbard’s *Vector Calculus,Linear Algebra And Differential Forms: A Unified Approach*, 3rd edition. It is an incredibly rich treasure trove of results, all presented with full rigor and with a striking amount of originality. What I love best about this book is that it balances rigorous mathematics with applications better then any textbook I’ve ever seen. It is a bit pricey, but now as a gift for Christmas, it’s perfect since it’ll be a lifetime investment for a serious student of mathematics.

It’s simply awesome, you’re wasting your time buying anything else. (I’m not saying there’s not any other good books on rigorous multivariable calculus-there certainly are. I just think any other book for a beginner is waste of money to **buy**. )

- Sum of reciprocals of numbers with certain terms omitted
- Proving vector calculus identity $\nabla \times (\mathbf a\times \mathbf b) =\cdots$ using Levi-Civita symbol
- Show this equality (The factorial as an alternate sum with binomial coefficients).
- Finding subgroups of a free group with a specific index
- Given $A^2$ where A is matrix, how find A?
- Cantor set: Lebesgue measure and uncountability
- Is self-adjointness really a property of an operator, or of an operator and an inner product?
- The sum of a polynomial over a boolean affine subcube
- Explanation of the binomial theorem and the associated Big O notation
- Poisson Integral is equal to 1
- Infinite Series $\sum\limits_{n=1}^\infty\frac{H_{2n+1}}{n^2}$
- Improper integral: $\int_1^{+\infty}\frac{\mathrm dx}{x(x+1)(x+2)\cdots(x+n)}$
- Proof of an alternative form of Fermat-Euler's theorem.
- Bijection from finite (closed) segment of real line to whole real line
- integral cohomology ring of real projective space