Advanced Calculus by Phil Dyke (auth.)

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By Phil Dyke (auth.)

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First of all let us redraw the diagram in critical position as this gives us criteria by which to judge whether the trolley can round the comer without tilting. 10 shows the trolley in this position and defines the angle 0 as that between the Ionger side of the trolley and a wall of the corridor. Using the notation of the figure, we Iabel the Ionger side of the trolley p and the comer that this side will just touch (in the limiting case) N. 10 The notation used. The trolley is in its limiting position.

For example, (I) is positive if = 0 If fxxfyy e but negative if tane fxx . In this case, the extremum is neither a maximum nor a minimum; it fxy is termed a saddle point. The name derives from the shape of the saddle familiar to the equestrian but here it encompasses many other shapes. For example, if (I) changed sign many times as e increased from 0 to 21t (like, say, 89), the shape of f(x, y) near such a point would resemble the central part of an old-fashioned jelly mould. _2 i[2fxy cosOsinO + f yy sin 2 9] which, in general, changes in sign and is thus a saddle point.

The gen- eralisation of the character of the extremum can also be made as follows: Using Taylor's Theorem for n variables 34 n F = f + ~ hf; + the first Summation is identically zero since itl ;! aaj X; n n ~ ~ = /; = h;hjij Ü for all i, a necessary condition for an extremum. If the quadratic form ~ 1 h;h jij is positive definite, then the extrem um is a minimum. On the other band, if the quadratic form is negative definite,; then the extremum is a maximum. The condition we require is that the quadratic form ;;1 L~ 1 a;hxj 1 is positive definite if and only if all > 0, a11 az1 all < 0, a11 az1 a1z > 0, a11 a1z a13 az1 azz az3 a3, a32 a33 a11 a1z a13 az1 azz az3 a31 a32 a33 > 0, ...

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