Volume of $E$ in $\mathbb{R}^4$ - AMM

  • MHB
  • Thread starter MountEvariste
  • Start date
  • Tags
    Volume
In summary, the volume of a set E in $\mathbb{R}^4$ can be defined as the measure of the 4-dimensional space occupied by the points in E and is calculated using the integral $\int\int\int\int_E dV$. The volume is directly related to the dimension of E, increasing as the dimension increases and decreasing as the dimension decreases. The volume can change under certain transformations, but is preserved under isometries. Real-world applications of calculating the volume in $\mathbb{R}^4$ include physics, engineering, and computer graphics.
  • #1
MountEvariste
87
0
Let $E$ be the set of all real $4$-tuples $(a, b, c, d)$ such that if $x, y \in \mathbb{ R}$, then:
$(ax+by)^2+(cx+dy)^2 \le x^2+y^2$.
Find the volume of $E$ in $\mathbb{R}^4$.​

Source: AMM.
 
Mathematics news on Phys.org
  • #2
Hint:

Show that $E$ is defined by the inequality $a^2 +b^2 +c^2 +d^2 \leqslant 1+(ad −bc)^2$ with $a^2 +c^2 \leqslant 1$ and $b^2 +d^2 \leqslant 1$.
 

FAQ: Volume of $E$ in $\mathbb{R}^4$ - AMM

What is the definition of volume in $\mathbb{R}^4$?

The volume of a set $E$ in $\mathbb{R}^4$ can be defined as the measure of the 4-dimensional space occupied by the points in $E$. This can be thought of as the amount of space that would be filled if $E$ was filled with water.

How is the volume of $E$ in $\mathbb{R}^4$ calculated?

The volume of $E$ in $\mathbb{R}^4$ can be calculated using the integral $\int\int\int\int_E dV$, where $dV$ represents an infinitesimal 4-dimensional volume element. This integral is typically evaluated using techniques from multivariable calculus.

What is the relationship between volume and dimension in $\mathbb{R}^4$?

In $\mathbb{R}^4$, the volume of a set $E$ is directly related to its 4-dimensional "size". As the dimension of $E$ increases, so does its volume. Similarly, as the dimension decreases, the volume decreases as well.

How does the volume of $E$ in $\mathbb{R}^4$ change under transformations?

The volume of $E$ in $\mathbb{R}^4$ can change under transformations such as translations, rotations, and dilations. However, the volume is preserved under certain types of transformations, such as isometries (rigid motions).

What are some real-world applications of calculating the volume of $E$ in $\mathbb{R}^4$?

The concept of volume in higher dimensions has many practical applications, such as in physics, engineering, and computer graphics. For example, calculating the volume of a 4-dimensional shape can help in understanding the behavior of a physical system with 4 spatial dimensions, or in creating 4-dimensional visualizations in computer graphics.

Similar threads

Replies
1
Views
826
Replies
5
Views
1K
Replies
2
Views
1K
Replies
27
Views
3K
Replies
7
Views
1K
Replies
3
Views
855
Replies
2
Views
1K
Replies
2
Views
545
Back
Top