Intereting Posts

Expression for $n$-th moment
n-th derivative of exponential function $\;e^{-f(x)}$
Equivalence of strong and weak induction
Why does the Fibonacci Series start with 0, 1?
Derivative of $ 4e^{xy ^ {y}} $
What are some easier books for studying martingale?
Can $\sin n$ get arbitrarily close to $1$ for $n\in\mathbb{N}?$
$2 \otimes_{R} 2 + x \otimes_{R} x$ is not a simple tensor in $I \otimes_R I$
What is the most efficient method to evaluate this indefinite integral?
Status of the classification of non-finitely generated abelian groups.
Integer factorization: What is the meaning of $d^2 – kc = e^2$
What is the purpose of Jordan Canonical Form?
Showing the polynomials form a Gröbner basis
Another property of the function $\phi : p \mapsto \int |f|^p d\mu$.
What is $\gcd(0,a)$, where a is a positive integer?

Kaplan’s Advanced Calculus defines the gradient of a function $f:\mathbb{R^n} \rightarrow \mathbb{R}$

as the $1 \times n$ *row vector* whose entries respectively contain the $n$ partial derivatives

of $f$. By this definition then, the gradient is just the Jacobian matrix of the transformation.

We also know that using the Riesz representation theorem, assuming $f$ is differentiable at the point

$x$, we can define the gradient as the unique vector $\nabla f$ such that

$$

df(x)(h) = \langle h, \nabla f(x) \rangle, \; h \in \mathbb{R}^n

$$

- Solve $\lim_{x\to +\infty}\frac{x^x}{(\lfloor x \rfloor)^{\lfloor x \rfloor }}$
- What is the use of hyperreal numbers?
- Continuous increasing function with different Dini derivatives at 0
- Convergence of $\sum_{n=1}^\infty\frac{n}{(n+1)!}$
- Showing that the norm of the canonical projection $X\to X/M$ is $1$
- A derivation of the Euler-Maclaurin formula?

Assuming we ignore the distinction between row vectors and column vectors, the former definition

follows easily from the latter. But, row vectors and column vectors are not the same things. So,

I have the following questions:

- Is the distinction here between row/column vectors important?
- If (1) is true, then how can we know from the second defintion that the vector

in question is a row vector and not a column vector?

- Inverse Partial Second Derivatives
- What does this $\asymp$ symbol mean? (subject: analytic number theory)
- Can any harmonic function on $\{z:0<|z|<1\}$ be extended to $z=0$?
- Continuity of multidimensional function: $f_1(x,y)=\frac{x^2y}{x^2+y^2}$ and $f_2(x,y)=\frac{2xy^3}{(x^2+y^2)^2}$
- What are the consequences if Axiom of Infinity is negated?
- Calculate center of mass multiple integrals
- Is there a step by step checklist to check if a multivariable limit exists and find its value?
- Volumes using triple integration
- Clarkson type inequality
- Removable singularity and laurent series

Yes, the distinction between row vectors and column vectors is important. On an arbitrary smooth manifold $M$, the derivative of a function $f : M \to \mathbb{R}$ at a point $p$ is a linear transformation $df_p : T_p(M) \to \mathbb{R}$; in other words, it’s a **cotangent** vector. In general the tangent space $T_p(M)$ does not come equipped with an inner product (this is an extra structure: see Riemannian manifold), so in general we cannot identify tangent vectors and cotangent vectors.

So on a general manifold one must distinguish between vector fields (families of tangent vectors) and differential $1$-forms (families of cotangent vectors). While $df$ is a differential form and exists for all $M$, $\nabla f$ can’t be sensibly defined unless $M$ has a Riemannian metric, and then it’s a vector field (and the identification between differential forms and vector fields now *depends on the metric*).

If one thinks of tangent vectors as column vectors, then $\nabla f$ ought to be a column vector, but the linear functional $\langle -, \nabla f \rangle$ ought to be a row vector. A major problem with working entirely in bases is that distinctions like these are frequently glossed over, and then when they become important students are very confused.

Some remarks about non-canonicity. The tangent space $T_p(V)$ to a vector space at any point can be *canonically* identified with $V$, so for vector spaces we don’t run into quite the same problems. If $V$ is an inner product space, then in the same way it automatically inherits the structure of a Riemannian manifold by the above identification. Finally, when people write $V = \mathbb{R}^n$ they frequently intend $\mathbb{R}^n$ to have the standard inner product with respect to the standard basis, and this equips $V$ with the structure of a Riemannian manifold.

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- Prime $p$ with $p^2+8$ prime
- If $T^2=TT^*$ then can i conclude that $T=T^*$?
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- Show that number of solutions satisfying $x^5=e$ is a multiple of 4?
- Proving that a compact subset of a Hausdorff space is closed
- Prove ${\large\int}_0^1\frac{\ln(1+8x)}{x^{2/3}\,(1-x)^{2/3}\,(1+8x)^{1/3}}dx=\frac{\ln3}{\pi\sqrt3}\Gamma^3\!\left(\tfrac13\right)$
- I read that ordinal numbers relate to length, while cardinal numbers relate to size. How do 'length' and 'size' differ?
- The asymptotic behaviour of $\sum_{k=1}^{n} k \log k$.
- $T(G)$ may not be a subgroup?
- Homotopy lifting property of $\mathbb{R} \to S^1$ in Hatcher