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21h
comment Abbreviating the definition of a tangent vector field?
I've seen $F: \mathbb{R}^n \rightarrow \mathbb{R}^n$ considered a vector field with the proviso that $F(x)$ is understood to be attached to $x$. Of course, the definition you write is a formalization of the informal slogan I give in this comment. Alternatively, you can define the tangent bundle and define a vector field as a section, but, I suspect that is further removed from your desire here.
2d
reviewed Approve Is an automorphism of the field of real numbers the identity map?
2d
comment Different Partial Derivatives Used in Curl in Proof of Stokes' Theorem
Glad to help, for what it's worth, you can also take a look at pages 390-392 of my notes where I go through the usual argument for Stokes' Theorem on a graph with a nice domain; supermath.info/CalculusIIIf2014.pdf
2d
comment Different Partial Derivatives Used in Curl in Proof of Stokes' Theorem
The answer is that you must pay attention to the context in which those calculations are given. It's all about the relations between $x,y$ and $z$ which depend on the context. It's not right to think about this just in terms of partial derivatives. The story is much more about the surface or boundary around which we are forming the surface or line integral.
Aug
31
reviewed Reject Determine the number of saddle points under specified conditions
Aug
30
comment A triple integral dancing in the unit cube
that's an improper integral. Or is it... make a common denominator.
Aug
30
comment $m \times n$ matrix gives rise to a well-defined map from $\mathbb{R}^n$ to $\mathbb{R}^m$?
well, that I don't know, I mean, certainly $A(BC)=(AB)C$ for matrix multiplication built with elements of a ring. Is there a way to build matrix multiplication over an object with less structure such that we lose associativity? That seems possible, but, I don't have an answer handy. On the other hand, matrix multiplication can represent operations which are not associative, for example, $[A,B]=AB-BA$ is well-known to fail associativity as measured by the Jacobi identity. Interesting question.
Aug
30
comment About a matrix identity.
I just noticed some homework problems about expressing the determinant in terms of the trace. I suspect, these might be related. Unfortunately, that book is at my office for now. Well, maybe this: math.stackexchange.com/questions/668374/…
Aug
30
comment $m \times n$ matrix gives rise to a well-defined map from $\mathbb{R}^n$ to $\mathbb{R}^m$?
I take your field and raise you a ring.
Aug
30
comment How do you find the null space of an inconsistent system?
probably, $(0,y,0)$ for any $y \in \mathbb{R}$ as those are the solutions to the homogenous equation. Or, it could be, $(0,y,0,0)$ from another viewpoint.
Aug
29
comment Is a circle in the xy plane considered a graph?
there are places where the term "graph" is reserved for those graphs which are attainable as the graphs of functions of a Cartesian variable. In such a context, a circle is not a "graph". Anyway, a circle is not a function period. Just like a line is not a function. The graph of a function $f(x)=mx+b$ is a line.
Aug
29
comment System of vector equations (in Minkowski space)
I remember the center of mass frame and the rest frame, but, this is a new one for me. Pretty answer.
Aug
27
comment Why is the Span of a subset of a linear space defined in such at way?
probably good to read en.wikipedia.org/wiki/Schauder_basis
Aug
27
reviewed Approve Is $P(n) = \frac{a n }{b}$ or $\frac{(a+1) n}{b + 1}$?
Aug
26
comment What mathematics topics pertain more towards applied mathematics?
Once the course is over and you have some time, maybe read something different on linear algebra. There are so many books which complement whatever you are using (notice I make this comment in complete ignorance of the specific text you are currently using)
Aug
26
comment What mathematics topics pertain more towards applied mathematics?
Raoul Bott: "80 percent of math is linear algebra" (rough quote)
Aug
26
comment Compute the operator norm of the linear transformation defined by the following matrix.
I guess $2$, maybe $3$ ?
Aug
25
comment Manifold that is not a Euclidean space
How about $S^1 \times S^1$ inside $\mathbb{C}^2$. This is a torus.
Aug
24
reviewed Approve uniform distribution, probability
Aug
23
comment Similar Matrices and their Jordan Canonical Forms
for matrices over a field which is not algebraically closed there is no guarantee that there are eigenvalues in the field. For example, over $\mathbb{R}$, most rotation matrices have pure imaginary eigenvalues. Consequently, there is no Jordan form. If you're working over $\mathbb{C}$ then what has been said here is fine. Moreover, if you really mean the real Jordan form for the real case then the equivalence also holds in that context. Bottom line, similar matrices have the same eigenvalues and geometric multiplicities hence the canonical forms match-up.