All Questions
78
questions
0
votes
0
answers
37
views
Is this calculus derivation process correct?
This is part of the economics romer model, finding the Lagrangian value
$$\mathcal{L}=\int_{i=0}^Ap(i)L(i)di-\lambda([\int_{i=0}^AL(i)^\phi di]^{\frac{1}{\phi}}-1)\\
s.t.\int_{i=0}^{A}L(i)^{\phi}di=1
$...
- 101
3
votes
0
answers
67
views
Problem 3-33 in "Calculus on Manifolds" by Michael Spivak. What is $g$? I cannot understand the author's intention for this problem at all.
Problem 3-33.
If $f:[a,b]\times [c,d]\to\mathbb{R}$ is continuous and $D_2f$ is continuous, define $F(x,y)=\int_a^x f(t,y)dt$.
(a) Find $D_1F$ and $D_2F$.
(b) If $G(x)=\int_a^{g(x)} f(t,x)dt$, find $G'...
- 1,138
2
votes
0
answers
104
views
Problem 3-32 in "Calculus on Manifolds" by Michael Spivak. What are "considerably weaker hypotheses"?
Problem 3-32.
Let $f:[a,b]\times [c,d]\to\mathbb{R}$ be continuous and suppose $D_2f$ is continuous. Define $F(y)=\int_a^b f(x,y) dx$. Prove Leibnitz's rule: $F'(y)=\int_a^b D_2f(x,y) dx$. Hint: $F(y)=...
- 1,138
0
votes
1
answer
51
views
Steepest Ascent Theorem
The temperature on the $Oxy$ plane is given by
f$(x, y) = 100−x^2 −2y^2$.
Assuming that an ant follows the path of steepest ascent of temperature, find and sketch
the path the ant follows if it starts ...
- 261
0
votes
1
answer
34
views
Does $\Delta (g(t, \cdot) *f) = \Delta g(t, \cdot) *f$ hold if $f$ is bounded continuous without compact support?
We consider the heat kernel
$$
g :\mathbb R_{>0} \times \mathbb R^d \to \mathbb R, (t, x) \mapsto \frac{1}{(4\pi t)^{d/2}} \exp \bigg ( - \frac{|x|^2}{4t} \bigg ).
$$
I would like to verify that
...
- 5,817
4
votes
1
answer
153
views
Divergence theorem when $\nabla \cdot \vec{F} = 0$
Calculate the flow of the vector field $$\mathbf{F}(x, y, z) = \frac{1}{(x^2 + y^2 + z^2)^{\frac{3}{2}}} (x, y, z)$$ out of a sphere with radius $10$ and center at the origin.
This what I did:
$$\frac{...
- 1,237
1
vote
2
answers
32
views
Determine the boundary $R$ of the area $D_2 $
Determine the boundary $R$ of the area $D_2 =$ {$(x, y) : x^2 + y^2 \leq 1, x > 0$}.
a) $R =$ {$x^2 + y^2 = 1, x \geq 0$} $\cup$ {($x, y) : x = 0, -1 < y < 1$}.
b) $R = ${$x^2 + y^2 = 1, x &...
- 1,237
1
vote
1
answer
102
views
How to calculate the surface area of an object?
Calculate the area of the surface Y given by the equation $z = x^2 + y^2 − 1$ when $z ≤ 0$
Here is my solution:
I got the answer correct but there is something I just did (not toally randomly but I ...
- 1,237
0
votes
1
answer
89
views
Stokes theorem for an intersection between a plane and a cylinder
Use Stokes' theorem to calculate the curve integral $\int_{\gamma} x dx + (y + z) dy + (y + z) dz$ where $\gamma$ is the intersection between the cylinder $x^2 + y^2 = 4$ and the plane $x + y + z = 0$....
- 1,237
0
votes
1
answer
55
views
Flux through a cylinder
Calculate the flow of the vector field $F(x, y, z) = (x, y, z)$ out
(away from the z-axis) through the cylinder surface $x^2 + y^2 = 4$,
when $0 ≤ z ≤ 1$, by stop at the surface and use the divergence
...
- 1,237
3
votes
3
answers
277
views
Find the volume by using triple integral
Calculate the volume of the contents of the bowl $K$, which is given by $x^2 + y^2 \leq z \leq 1$.
$$x^2+y^2\leq z\leq 1$$
$$x^2+y^2=1\iff y=\pm\sqrt{1-x^2}$$
$$-1\leq x\leq 1$$
$$-\sqrt{1-x^2}\leq ...
- 1,237
1
vote
1
answer
49
views
Triple integral over a half hemisphere?
Calculate the integral $\displaystyle \iiint_K (x + 2) d{x} d{y} d{z}$ when $K$ is
given by the inequalities $x^2 + y^2 + z^2 ≤ 1$ and $z ≥ 0$.
But the correct answer is $\frac{4\pi}3$ and they gave ...
- 1,237
4
votes
2
answers
209
views
Calculate the integral over the half ellipse
Calculate the integral $\displaystyle \iint_E x^2 dA$ where E is the half ellipse given by the inequalities $x^2 + 2y^2 \le 2$ and $y \ge 0$.
This is what I did:
$$\begin {align}\iint_E x^2 dA &=\...
- 1,237
0
votes
1
answer
84
views
How to integrate the canonical form of the 2nd order PDE?
Given the 2nd order linear PDE
\begin{align}
x^2u_{xx}-2xtu_{xt}+t^2u_{tt}+xu_x+tu_t &=0, & &x>0,\ t \in \mathbb{R} \tag 1
\end{align}
It is parabolic. I used the following ...
2
votes
1
answer
59
views
Calculate $\int_{0}^{\infty}\frac{1}{(x^2 + t^2)^n}dx$ for $t > 0, n \ge 1$
The problem is as follows:
Show that for all $t > 0, n \ge 1$,
$$
\int_{0}^{\infty}\frac{1}{(x^2 + t^2)^n}dx = {2n-2 \choose n -1}\frac{\pi}{(2t)^{2n -1}}
$$
What I have so far:
Let $f(t) = \int_{0}...
- 157