A Jacobian Worked out example


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Hyperbolic Paraboloid

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Graphs and Level curves of functions of Two variables Definitions The set of points (x,y) in the plane where a function f (x,y) has a constant value f(x,y)=c is called a level curve of f The set of all points (x,y,f (x,y)) in space, for (x,y) in the domain of f, is called the graph of f The graph of f is called the surface z=f (x,y) The equation of the following hyperbolic paraboloid is `z=50+x^2-y^2`
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 Four identical solid spheres of mass 'm' and radius 'a' are placed as shown Find the moment of inertia of the system through A and axis parallel to DB Since the size of sphere is small it can be considered as a point object and moment of inertia about the given line can be calculated as follows. DE and BF can be calculated from respective triangles and CA and EF are the diagonals of the square. A is the mid point of EF. `I=M\timesK^2` `I=m\times 0^2 + m\times BF^2 + m\times CA^2 + m\times DE^2` =`m\timesl^2/2+m\times(sqrt(2) l)^2+m\times l^2/2` =`3ml^2`
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