Question

Prove that $\int_0^\pi \ln(1+\alpha \cos x)\, dx= \pi \ln\left(\frac{1+\sqrt{1-\alpha^2}}{2}\right)$

score 206 · Answer 1 · 2026-04-16 22:03:09

206

Views

Prove that $\int_0^\pi \ln(1+\alpha \cos x)\, dx= \pi \ln\left(\frac{1+\sqrt{1-\alpha^2}}{2}\right)$

Published on 16 Apr 2026 - 22:03

score 33 · Answer 2 · 2026-04-13 03:05:28

33

Views

Any idea on how to find a upper bound for a limit in R^2?

Published on 13 Apr 2026 - 3:05

score 743 · Answer 3 · 2026-04-16 08:52:54

743

Views

Q: Find the envelope of a family of straight lines whereby distance between x and y intercept is a constant S.

Published on 16 Apr 2026 - 8:52

score 138 · Answer 4 · 2026-04-17 10:26:10

138

Views

$f(x) = (x_1-x_2^2)(x_1-\frac{1}{2}x_2^2)$, verify $\overline{x} = (0,0)^t$ is local min of $\phi(\lambda) = f(\overline{x}+\lambda d)$ but not of $f$

Published on 17 Apr 2026 - 10:26

score 187 · Answer 5 · 2026-04-17 10:29:05

187

Views

Show that $\int_0^2 \int_0^2 \frac{x}{1+\ln(x^2y^2)} \,\mathrm{d}x \,\mathrm{d}y \leq 4$

Published on 17 Apr 2026 - 10:29

score 1.1k · Answer 6 · 2026-04-12 08:28:03

1.1k

Views

Principal curvatures of surface of revolution

Published on 12 Apr 2026 - 8:28

score 112 · Answer 7 · 2026-04-16 10:39:19

112

Views

Find a descent direction $d^k$ orthogonal to direction $d^{k-1}$ which is linear comb. of $d^{k-1}$ and $\nabla f(x_k)$

Published on 16 Apr 2026 - 10:39

score 131 · Answer 8 · 2026-04-16 10:51:00

131

Views

Find the $\lim_{(x,y)\to (0,0)} \frac {x^2+y^4}{|x|+|y|} $

Published on 16 Apr 2026 - 10:51

score 56 · Answer 9 · 2026-04-11 14:34:52

56

Views

Question about limits.

Published on 11 Apr 2026 - 14:34

score 455 · Answer 10 · 2026-04-12 14:29:48

455

Views

Is there a smooth surjective function $f:\mathbb{R}^2 \to \mathbb{R}^3$?

Published on 12 Apr 2026 - 14:29

score 87 · Answer 11 · 2026-04-12 11:29:47

87

Views

Partial chain rule in multi-dimensional case

Published on 12 Apr 2026 - 11:29

score 43 · Answer 12 · 2026-04-15 13:56:38

43

Views

Let $A: \Bbb{R}^{+}\to M_{n\times n}(\Bbb{R})$, then $A(t)$ is continuous if and only if $a_{ij}(t)$ is continuous, $\forall\;i,j=1,\cdots,n$

Published on 15 Apr 2026 - 13:56

score 145 · Answer 13 · 2026-04-15 23:43:20

145

Views

$L= \left( \frac{1}{r\sin\theta} \frac{\partial}{\partial \phi} \hat{\phi} - \frac{1}{r}\frac{\partial }{\partial \theta} \hat{\theta} \right) $

Published on 15 Apr 2026 - 23:43

score 64 · Answer 14 · 2026-05-04 22:05:32

64

Views

If $x\mapsto F(x)=\int^{x}_{a}f(s)ds$, then $F$ is $C^1$ and $F'(x)=f(x),\;\;\forall\;x\in[a,b]$

Published on 04 May 2026 - 22:05

score 97 · Answer 15 · 2026-04-10 23:34:51

97

Views

Multi-variable chain rule with multi-variable functions as arguments

Published on 10 Apr 2026 - 23:34

List Question

Prove that $\int_0^\pi \ln(1+\alpha \cos x)\, dx= \pi \ln\left(\frac{1+\sqrt{1-\alpha^2}}{2}\right)$

Any idea on how to find a upper bound for a limit in R^2?

Q: Find the envelope of a family of straight lines whereby distance between x and y intercept is a constant S.

$f(x) = (x_1-x_2^2)(x_1-\frac{1}{2}x_2^2)$, verify $\overline{x} = (0,0)^t$ is local min of $\phi(\lambda) = f(\overline{x}+\lambda d)$ but not of $f$

Show that $\int_0^2 \int_0^2 \frac{x}{1+\ln(x^2y^2)} \,\mathrm{d}x \,\mathrm{d}y \leq 4$

Principal curvatures of surface of revolution

Find a descent direction $d^k$ orthogonal to direction $d^{k-1}$ which is linear comb. of $d^{k-1}$ and $\nabla f(x_k)$

Find the $\lim_{(x,y)\to (0,0)} \frac {x^2+y^4}{|x|+|y|} $

Question about limits.

Is there a smooth surjective function $f:\mathbb{R}^2 \to \mathbb{R}^3$?

Partial chain rule in multi-dimensional case

Let $A: \Bbb{R}^{+}\to M_{n\times n}(\Bbb{R})$, then $A(t)$ is continuous if and only if $a_{ij}(t)$ is continuous, $\forall\;i,j=1,\cdots,n$

$L= \left( \frac{1}{r\sin\theta} \frac{\partial}{\partial \phi} \hat{\phi} - \frac{1}{r}\frac{\partial }{\partial \theta} \hat{\theta} \right) $

If $x\mapsto F(x)=\int^{x}_{a}f(s)ds$, then $F$ is $C^1$ and $F'(x)=f(x),\;\;\forall\;x\in[a,b]$

Multi-variable chain rule with multi-variable functions as arguments

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