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Results tagged with calculus
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user 282830
Calculus is the branch of mathematics which deals with the study of rate of change of quantities. This is usually divided into differential calculus and integral calculus which are concerned with derivatives and integrals respectively. DO NOT USE THIS TAG just because your question makes use of calculus.
0
votes
Accepted
Laplacian of exponential of vector norm (for heat equation)
The solution is
$$\tag{1}
V(\vec{u};t) = N \exp{\frac{|\vec{u}-\vec{u}_0|^2}{4 D t}}
$$
Where $N$ is the normalization constant $N = \frac{1}{(6\pi D t)^{3/2}}$
To do the Laplacian of Eq.(1), you m …
- 4,314
3
votes
Finding the centre of mass in polar coords with double integrals
The 2-d vector $\vec{r}$
$$
\vec{r} = \hat{x} x + \hat{y} y \ne \hat{r} r + \hat{\phi} \phi.
$$
Instead, the vector in polar coordinate:
$$
\vec{r} = \hat{r} r.
$$
Therefore, the center of mass $ …
- 4,314
4
votes
Is it correct to equate the same thermodynamic potential with different variable dependencies?
They simply the calculus chain-rule and change variables. Nothing wrong with that. You don't need a thermodynamic property to validate these equations. It is rigorous mathematical relation. …
- 4,314
1
vote
Accepted
Toss of a ball using a double integral
Put it in this way for easy to grab each step:
$$ \tag{1}
x(t) = x(0) + \int_0^t v(\tau) d\tau.
$$
and
$$ \tag{2}
v(\tau) = v(0) + \int_0^\tau a(\xi) d\xi.
$$
These are basic kinematic relatio …
- 4,314
3
votes
Accepted
Infinitesimals meaning in this context
The notation $M(dt)$ or $v(dt)$ is not acceptable. Typical notations are like:
At the time $t$:
mass of rocket $M(t)$
velocity of rocket $\vec{v}(t)$
Then at the time $t+dt$
mass of rocket $M(t+dt …
- 4,314