Warning: Invalid argument supplied for foreach() in /var/www/kartinki1/data/www/3.1onega.ru/ftrans/proxy.php on line 19

Warning: array_diff(): Argument #2 is not an array in /var/www/kartinki1/data/www/3.1onega.ru/ftrans/proxy.php on line 29

Warning: Invalid argument supplied for foreach() in /var/www/kartinki1/data/www/3.1onega.ru/ftrans/proxy.php on line 19

Warning: array_diff(): Argument #1 is not an array in /var/www/kartinki1/data/www/3.1onega.ru/ftrans/proxy.php on line 29

Warning: Invalid argument supplied for foreach() in /var/www/kartinki1/data/www/3.1onega.ru/ftrans/proxy.php on line 19

Warning: array_diff(): Argument #1 is not an array in /var/www/kartinki1/data/www/3.1onega.ru/ftrans/proxy.php on line 29
Snow globe with his hands

Snow globe with his hands

Date: 16.10.2018, 22:00 / View: 31532

Wikipedia, the free encyclopedia Go to navigation Go to search

This term has other meanings, see Shar (meanings). The surface of the ball is a sphere
r is the radius of the ball

A ball is a geometric body; the set of all points of space that are at a distance from the center is not more than the specified one. This distance is called the radius of the ball. The ball is formed by rotating a semicircle near its fixed diameter. This diameter is called the axis of the ball, and both ends of the specified diameter - the poles of the ball. The surface of the ball is called a sphere: a closed ball includes this sphere, the open ball excludes. Content

If the cutting plane passes through the center of the ball, then the section of the ball is called a large circle. Other flat sections of the ball are called small circles. The area of ​​these sections is calculated by the formula πR².

Surface area S {\ displaystyle S} S and volume V {\ displaystyle V} < img src = "https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" alt = "V"> ball of radius r {\ displaystyle r}  r (and diameter d = 2 r {\ displaystyle d = 2r}  d = 2r ) are defined by the formulas: S = 4 π r 2 {\ displaystyle S = \ 4 \ pi r ^ {2}}  S = \ 4 \ pi r ^ {2} S = π d 2 {\ displaystyle S = \ \ pi d ^ {2}}  S = \ pi d ^ {2} V = 4 3 π r 3 {\ displaystyle V = {\ frac {4} {3} } \ pi r ^ {3}} V = {\ frac {4} {3} } \ pi r ^ {3} V = π d 3 6 {\ displaystyle V = {\ frac {\ pi d ^ {3}} {6}}  V = {\ frac {\ pi d ^ {3}} {6}}

The concept of a ball in a metric space naturally generalizes the concept of a ball in Euclidean geometry.

Let the given space be (X, ρ) {\ displaystyle (X, \ rho)} . Then A ball (or an open ball) with a center at x 0 ∈ X {\ displaystyle x_ {0} \ in X}  x_ {0} \ in X and radius r> 0 {\ displaystyle r> 0}  r> 0 is the set B r (x 0) = {x ∈ X ∣ ρ (x, x 0) A closed ball with a center at x 0 {\ displaystyle x_ {0}} = and radius r {\ displaystyle r}  r is the set D r (x 0) = {x ∈ X ∣ ρ (x, x 0) ⩽ r}. {\ displaystyle D_ {r} (x_ {0}) = \ {x \ in X \ mid \ rho (x, x_ {0}) \ leqslant r \}.}  D_ {r} (x_ {0}) = \ {x \ in X \ mid \ rho (x, x_ {0}) \ leqslant r \}. Comments [edit | edit code]

Sphere radius r {\ displaystyle r} r with center x 0 {{displaystyle x 0}} x_ {0} is also called r {\ displaystyle r} r - by the neighborhood of the point x 0 {\ displaystyle x_ {0}} x x {0} . The ball is an open set in the topology, generated by the metric ρ {\ displaystyle \ rho} . A closed ball is closed by a set in the topology generated by the metric ρ {\ displaystyle \ rho} X are its base. Obviously, B r (x 0) ⊂ D r (x 0) {\ displaystyle B_ {r} (x_ {0}) \ subset D_ {r} (x_ {0})}  B_ {r} (x_ {0}) \ subset D_ {r} (x_ {0}) . However, generally speaking, the closure of an open ball may not coincide with a closed ball: B r (x 0) ¯ D r (x 0). {\ displaystyle {\ overline {B_ {r} (x_ {0})}} \ neq D_ {r} (x_ {0}).}  \ overline {B_ {r} (x_ {0})} \ neq D_ {r} (x_ {0}). For example: let (X, ρ) {\ displaystyle (X, \ rho)} (X, \ rho) discrete metric space, and X {\ displaystyle X} X consists of more than two points . Then for any x ∈ X {\ displaystyle x \ in X} x \ in X we have: B 1 (x) = {x}, B 1 (x) ¯ = {x}, D 1 (x) = X. {\ displaystyle B_ {1} (x) = \ {x \}, \; {\ overline {B_ {1} (x)}} = \ {x \}, \; D_ {1} (x) = X .} B_ {1} (x) = \ {x \}, \; \ overline {B_ {1} (x)} = \ {x \}, \; D_ {1} (x) = X. Let R d {\ displaystyle \ mathbb {R} ^ {d}} \ mathbb {R} ^ d is a Euclidean space with regular Euclidean distance. Then if d = 1 {\ displaystyle d = 1} d = 1 (space - line), then B r (x 0) = {x ∈ R ∣ | x - x 0 | D r (x 0) = {x ∈ R ∣ | x - x 0 | ≤ r} = [x 0 - r, x 0 + r]. {\ displaystyle D_ {r} (x_ {0}) = \ {x \ in {\ mathbb {R}} \ mid | x-x_ {0} | \ leq r \} = \ left [x_ {0} - {r}, x_ {0} + {r} \ right].} D_ {r } (x_ {0}) = \ {x \ in {\ mathbb R} \ mid | x-x_ {0} | \ leq r \} = \ left [x_ {0} - {r}, x_ {0} + {r} \ right]. is an open and closed segment, respectively. if d = 2 {\ displaystyle d = 2} d = 2 (space - plane ), then B r ((x 0, y 0)) = {(x, y) ∈ R 2 ∣ (x - x 0) 2 + (y - y 0) 2 D r ((x 0, y 0)) = {(x, y) ∈ R 2 ∣ (x - x 0) 2 + (y - y 0) 2 ≤ r} {\ displaystyle D_ {r} ((x_ {0}, y_ {0})) = \ left \ {(x, y) \ in \ mathbb {R} ^ {2} \ mid {\ sqrt {(x-x_ {0}) ^ {2} + (y-y_ {0}) ^ {2} }} \ leq r \ right \}} D_ {r} ((x_ {0} , y_ {0})) = \ le ft \ {(x, y) \ in {\ mathbb {R}} ^ {2} \ mid {\ sqrt {(x-x_ {0}) ^ {2} + (y-y_ {0}) ^ { 2}}} \ leq r \ right \} - open and closed disk, respectively. if d = 3 {\ displaystyle d = 3}  d = 3 , then B r ((x 0, y 0, z 0)) = {(x, y, z) ∈ R 3 (x - x 0) 2 + (y - y 0) 2 + (z - z 0) 2 D r ((x 0, y 0, z 0)) = {(x, y, z) ∈ R 3 ∣ (x - x 0) 2 + (y - y 0) 2 + (z - z 0) 2 ≤ r} {\ displaystyle D_ {r} ((x_ {0}, y_ {0}, z_ {0})) = \ left \ {(x, y, z) \ in \ mathbb {R} ^ {3 } \ mid {\ sqrt {(x-x_ {0}) ^ {2} + (y-y_ {0}) ^ {2} + ( z-z_ {0}) ^ {2}}} \ leq r \ right \}} D_ {r} ((x_ {0}, y_ {0}, z_ {0})) = \ left \ {(x, y, z) \ in {\ mathbb {R}} ^ {3} \ mid {\ sqrt {(x-x_ {0}) ^ {2} + (y-y_ {0}) ^ {2} + (z-z_ {0}) ^ {2}} \ leq r \ right \ } - open and closed stereometric ball, respectively. In other metrics, the ball may have a different geometric shape. For example, let's define in the Euclidean space Rd {\ displaystyle \ mathbb {R} ^ {d}} The "\ mathbb {R} ^ d"> metric is as follows: ρ (x, y) = ∑ i = 1 d ‖ xi - yi ‖, x = (x 1, ..., xd), y = (y 1, ..., yd) ⊤ ∈ R d. {\ displaystyle \ rho (x, y) = \ sum \ limits _ {i = 1} ^ {d} \ | x_ {i} -y_ {i} \ |, \ quad x = (x_ {1}, \ ldots, x_ {d}) ^ {\ top}, y = (y_ {1}, \ ldots, y_ {d}) ^ {\ top} \ in \ mathbb {R} ^ {d}.} \ rho (x, y) = \ sum \ limits _ {{i = 1}} ^ {d } \ | x_ {i} -y_ {i} \ |, \ quad x = (x_ {1}, \ ldots, x_ {d}) ^ {{\ top}}, y = (y_ {1}, \ ldots, y_ {d}) ^ {{\ top}} \ in {\ mathbb {R}} ^ {d}. Then if d = 2 {\ displaystyle d = 2}  d = 2 , then U r (x 0) {\ displaystyle U_ {r} (x_ {0})} < img src = "https://wikimedia.org/api/rest_v1/media/math/render/svg/c311b86c6d6cada2f029fb62768fcdf4e120d88f" alt = "U_ {r} (x_ {0})"> is an open square centered at x 0 {\ displaystyle x_ {0}} x_ {0} and side mi length 2 {\ displaystyle {\ sqrt {2}}} {\ sqrt {2} } located diagonally to the coordinate axes. if d = 3 {\ displaystyle d = 3} d = 3, then, then (x 0) {\ displaystyle U_ {r} (x_ {0})} Ufc2f029fb62768fcdf4e120d88f is an open three-dimensional octahedron.

Source - https://ru.wikipedia.org/w/index.php?title=Shar&oldid=94085009


Related news:


Homemade car speakers
Sliding door to dressing room with their hands
The gas generator car based your hands
Truck for the boat motor with his own hands


Content parser: contentBingRss |
Total Parsers:1
ReCache | DelPage
Memory used: 285.8KB of 2.75MB
Render time: 9.605 sec., Version: 3.5.5