# All Questions

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### directional derivative problem

for a point M(4,1) and a function $z = x * y^2 - (x^2/y^3)$ I was tasked with finding a directional derivative in the direction which creates a 30 degree angle with the X axis....I find it a little ...
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### Example 5, Sec. 23 in Munkres' TOPOLOGY, 2nd edition: What is the closure of this set?

What is the closure in $\mathbb{R}^2$ of the set $$\left\{ \ x \times y \ \in \mathbb{R} \times \mathbb{R} \ \colon \ x > 0, \ y = \frac{1}{x} \ \right\}?$$ I know that each point of the set is ...
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### I want to learn mathematics to extend myself.

I am a middle school student that is highly fascinated by mathematics and its elegance. However, I have found that to appreciate a lot of mathematics a lot of knowledge is required. Currently, I can ...
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### Trace in Einstein notation

I know quite well what the trace of a matrix is; however, I am not quite sure I understand the meaning of the 'trace' concept when applied to tensors. I would be very grateful to you if: 1) You could ...
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### modules operation with different moduli

What value it will give $((a^b \bmod n_1)\cdot(c^d \bmod n_2))^f$; it $f$ will multiplied with both the power. There is some property of modulo operation with different moduli.
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### Which of the following is/are always correct.

Let $A$ be a $4\times7$ real matrix and $B$ be a $7\times4$ real matrix such that $AB=I_4$, where $I_4$ is the $4\times4$ identity matrix. Which of the following are is/are always true ...
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### Is every Pisot-like integer the product of a Pisot integer and a root of unity?

For lack of better terminology, let's call an algebraic number $\beta$ Pisot-like if $|\beta| > 1$ and all its conjugates lie inside the complex unit circle (here $|\cdot|$ is the usual absolute ...
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### Induction proofs - methodology and examples

Forenote: I have seen many induction related questions, and very often the problem lies within the OP's lack of a proper methodology (or style) in writing the proof whereas the answers focus on the ...
I've seen that the depth of the Cantor/Kaltofen algorithm is in $O(\log n)$. Are the operations for this complexity undifferentiated ? Or this complexity is in terms of multiplications only ?