Morse Mystery Cartoon | Science Experimental Animation | Morse Experiment Cartoon Animation | Morse Theory
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Morse Mystery Cartoon | Science Experimental Animation | Morse Experiment Cartoon Animation
Consider, for purposes of illustration, a mountainous landscape {\displaystyle M.}{\displaystyle M.} If {\displaystyle f}f is the function {\displaystyle M\to \mathbb {R} }{\displaystyle M\to \mathbb {R} } sending each point to its elevation, then the inverse image of a point in {\displaystyle \mathbb {R} }\mathbb {R} is a contour line (more generally, a level set). Each connected component of a contour line is either a point, a simple closed curve, or a closed curve with a double point. Contour lines may also have points of higher order (triple points, etc.), but these are unstable and may be removed by a slight deformation of the landscape. Double points in contour lines occur at saddle points, or passes. Saddle points are points where the surrounding landscape curves up in one direction and down in the other.
Contour lines around a saddle point
Imagine flooding this landscape with water. Then, the region covered by water when the water reaches an elevation of {\displaystyle a}a is {\displaystyle f^{-1}(-\infty ,a]}{\displaystyle f^{-1}(-\infty ,a]}, or the points with elevation less than or equal to {\displaystyle a.}a. Consider how the topology of this region changes as the water rises. It appears, intuitively, that it does not change except when {\displaystyle a}a passes the height of a critical point; that is, a point where the gradient of {\displaystyle f}f is {\displaystyle 0}{\displaystyle 0} (that is the Jacobian matrix acting as a linear map from the tangent space at that point to the tangent space at its image under the map {\displaystyle f}f does not have maximal rank). In other words, it does not change except when the water either (1) starts filling a basin, (2) covers a saddle (a mountain pass), or (3) submerges a peak.
The torus
To each of these three types of critical points – basins, passes, and peaks (also called minima, saddles, and maxima) – one associates a number called the index. Intuitively speaking, the index of a critical point {\displaystyle b}b is the number of independent directions around {\displaystyle b}b in which {\displaystyle f}f decreases. More precisely the index of a non-degenerate critical point {\displaystyle b}b of {\displaystyle f}f is the dimension of the largest subspace of the tangent space to {\displaystyle M}M at {\displaystyle b}b on which the Hessian of {\displaystyle f}f is negative definite. Therefore, the indices of basins, passes, and peaks are {\displaystyle 0,1,}{\displaystyle 0,1,} and {\displaystyle 2,}{\displaystyle 2,} respectively.
Consider, for purposes of illustration, a mountainous landscape {\displaystyle M.}{\displaystyle M.} If {\displaystyle f}f is the function {\displaystyle M\to \mathbb {R} }{\displaystyle M\to \mathbb {R} } sending each point to its elevation, then the inverse image of a point in {\displaystyle \mathbb {R} }\mathbb {R} is a contour line (more generally, a level set). Each connected component of a contour line is either a point, a simple closed curve, or a closed curve with a double point. Contour lines may also have points of higher order (triple points, etc.), but these are unstable and may be removed by a slight deformation of the landscape. Double points in contour lines occur at saddle points, or passes. Saddle points are points where the surrounding landscape curves up in one direction and down in the other.
Contour lines around a saddle point
Imagine flooding this landscape with water. Then, the region covered by water when the water reaches an elevation of {\displaystyle a}a is {\displaystyle f^{-1}(-\infty ,a]}{\displaystyle f^{-1}(-\infty ,a]}, or the points with elevation less than or equal to {\displaystyle a.}a. Consider how the topology of this region changes as the water rises. It appears, intuitively, that it does not change except when {\displaystyle a}a passes the height of a critical point; that is, a point where the gradient of {\displaystyle f}f is {\displaystyle 0}{\displaystyle 0} (that is the Jacobian matrix acting as a linear map from the tangent space at that point to the tangent space at its image under the map {\displaystyle f}f does not have maximal rank). In other words, it does not change except when the water either (1) starts filling a basin, (2) covers a saddle (a mountain pass), or (3) submerges a peak.
The torus
To each of these three types of critical points – basins, passes, and peaks (also called minima, saddles, and maxima) – one associates a number called the index. Intuitively speaking, the index of a critical point {\displaystyle b}b is the number of independent directions around {\displaystyle b}b in which {\displaystyle f}f decreases. More precisely the index of a non-degenerate critical point {\displaystyle b}b of {\displaystyle f}f is the dimension of the largest subspace of the tangent space to {\displaystyle M}M at {\displaystyle b}b on which the Hessian of {\displaystyle f}f is negative definite. Therefore, the indices of basins, passes, and peaks are {\displaystyle 0,1,}{\displaystyle 0,1,} and {\displaystyle 2,}{\displaystyle 2,} respectively.