Rax Adaam
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Taylor expansion of $\arccos(1-x)$ around $x=0$ to two terms
2 votes

Although addressed indirectly in the other answers here, it seems relevant to point out that, strictly speaking, the resulting series expansion in all cases (so far) is -not- a Taylor series. The ...

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Is it necessary to include "for all" when using set builder notation to define the image set?
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1 votes

Based on common conventions for set builder notation, I believe the "for all" or "any" is implied. For example, to refer to the set $A$ of all real numbers from 0 to 5, it is ...

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curl(fF) with Einstein Summation Notation
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1 votes

Recall that $\vec{a} \times \vec{b} = - \, \vec{b} \times \vec{a}$. Now consider each, using your definitions (I have included the unit vector $\hat{e}_i$ explicitly, simply for additional clarity; ...

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Does the "field" over which a vector space is defined have to be a Field?
0 votes

While the other answers (and comments) implicitly address the question stated in the title of the OP, I thought it may be useful to include an explicit answer, as well. Does the “field” over which a ...

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$-(k\vec{v})$ Vs. $-k\vec{v}$, for $k \in \mathbb{R}$ and $\vec{v} \in \mathbb{S}$ : Additive Inverse Vs. Scalar Multiplication by (-1)
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0 votes

A summary of the conclusion drawn from the comments (thanks to @saulspatz) Given that we write $k \odot \vec{v} = k\vec{v}$, in the case of the scalar $-1$, this would be either $$-1 \odot \vec{v} = -...

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Non-analytic smooth function
0 votes

The power series $$\sum_{n=0}^{\infty} 0 \, x^n = 0 + 0x + 0x^2 + 0x^3 + \dots$$ clearly converges to zero for any value of $x$. That is, if we call this power series $g(x)$, then we can say $g(x) = 0$...

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What is the Lagrange remainder in a Taylor series expansion
0 votes

Taking your comment about “any remainder” in mind, BCLC’s comment is effectively the answer you seem to be looking for, i.e. the remainder $R_n(x)$ is simply the difference between the $n^{th}$ order ...

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Correct algorithm for finding regions of increasing / decreasing and the relationship to critical points.
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0 votes

After further reflection, thanks to saulspatz's comment, a better formulation occurred to me, that seems to avoid the imprecision (and especially the overly inclusive language) of the formulation ...

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Do we have to take the absolute value of the jacobian ONLY if it is a number?
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I'm adding an additional answer, in case the issue is actually with the integration of an absolute value, recall the definition of the absolute value: $\begin{eqnarray*} \left|-2x+y\right| \quad &...

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