Intereting Posts

$\epsilon$-$\delta$ proof that $f(x) = x^3 /(x^2+y^2)$, $(x,y) \ne (0,0)$, is continuous at $(0,0)$
Can the square of a proper ideal be equal to the ideal?
How can you pick the odd marble by 3 steps in this case?
Group theory text
Are regular languages necessarily deterministic context-free languages?
Fermat numbers are coprime
Diffeomorphisms and Lipschitz Condition
Show that a sequence of functions convergent pointwise to $\chi_{\mathbb Q}$ does not exist
Prove that $\frac{(n!)!}{(n!)^{(n-1)!}} $ is always an integer.
Algorithm for comparing the size of extremely large numbers
Configuration scheme of $n$ points
In what sense are math axioms true?
Infinite product of sinc functions
How to arrive at Ramanujan nested radical identity
Let $A, B$ be sets. Show that $\mathcal P(A ∩ B) = \mathcal P(A) ∩ \mathcal P(B)$.

From the definitions I’m reading between the two:

The gradient vector field is defined by its construction: gradient of a scalar (or real) function generally over two or more variables.

The conservative vector field is defined by the common characteristic of every curve in this field: only the endpoints matter, not the path.

- Definition of the total derivative.
- Two paths that show that $\frac{x-y}{x^2 + y^2}$ has no limit when $(x,y) \rightarrow (0,0)$
- Uniform convergence in a proof of a property of mollifiers in Evans's Partial Differential Equations
- Geometric interpretation of mixed partial derivatives?
- Explain $\iint \mathrm dx\,\mathrm dy = \iint r \,\mathrm \,d\alpha\,\mathrm dr$
- Is there limit $ \lim_{(x,y) \to (0,0)} \frac{x^3}{x^2 + y^2}$?

Interpretation wise in traditional multi-variable calculus view, these two type of fields sound exactly the same. I’ve had some difficulty trying to pinpoint which one is more abstract or specialized, much less their difference.

According to Wikipedia (I may have committed blasphemy):

Conservative vector fields and the gradient theorem

The gradient of a function is called a gradient field. A (continuous)

gradient field is always a conservative vector field: its line

integral along any path depends only on the endpoints of the path, and

can be evaluated by the gradient theorem (the fundamental theorem of

calculus for line integrals). Conversely, a (continuous) conservative

vector field is always the gradient of a function.

Obviously this doesn’t help trying to understand the difference, if any.

- $z=f\left(xy,\, \frac y x \right)$ Differentiate
- Express partial derivatives of second order (and the Laplacian) in polar coordinates
- Multivariable Calculus books similar to “Advanced Calculus of Several Variables” by C.H. Edwards
- Equation of first variation for a flow
- If a nonnegative function of $x_1,\dots,x_n$ can be written as $\sum g_k(x_k)$, then the summands can be taken nonnegative
- Spherical coordinates for sphere with centre $\neq 0$
- determine whether $f(x, y) = \frac{xy^3}{x^2 + y^4}$ is differentiable at $(0, 0)$.
- Laplace Operator in $3D$
- Area form for $M^2 \subseteq \Bbb R^4$
- Proving vector calculus identity $\nabla \times (\mathbf a\times \mathbf b) =\cdots$ using Levi-Civita symbol

They are equivalent. Typically you define a conservative vector field $\mathbf{v}$ as one where there exists a scalar field $\phi$ such that $\mathbf{v} = \nabla \phi$. Subsequently you can use the gradient theorem to prove that an integral along a path only depends on the endpoints. The converse is equivalent so you can define it as you have also.

- Give an example of non-normal subspace of a normal space.
- Is $G/pG$ is a $p$-group?
- possible signatures of bilinear form on subspaces
- How to find basis for intersection of two vector spaces in $\mathbb{R}^n$
- Homology of the Klein Bottle
- Product of all elements in an odd finite abelian group is 1
- The Euler characteristic & a cube with holes?
- distance between two eigen vectors corresponding to two different matrices in a normed space
- Can the determinant of an integer matrix with a given row be any multiple of the gcd of that row?
- If $a,b$ are positive integers such that $\gcd(a,b)=1$, then show that $\gcd(a+b, a-b)=1$ or $2$. and $\gcd(a^2+b^2, a^2-b^2)=1$ or $2 $
- Sum and Product of two transcendental numbers cannot be simultaneously algebraic
- Ideals in $\mathbb{Z}$ with three generators (and not with two)
- Two disjoint spanning trees, spanning subgraph with all even degrees
- Error solving “stars and bars” type problem
- (Question) on Time-dependent Sobolev spaces for evolution equations