a 3D solid is bounded by 2 paraboloids. The binding condition in cartesian coordinates is
-1+(x[sup:29hv4ru9]2[/sup:29hv4ru9]+y[sup:29hv4ru9]2[/sup:29hv4ru9]) < 2z < 1-(x[sup:29hv4ru9]2[/sup:29hv4ru9]+y[sup:29hv4ru9]2[/sup:29hv4ru9])
a) rewrite the binding condition in parabolic coordinates
b) using parabolic coordinates and the given metric tensor, find the volume of the solid
using x=stcos(p) y= stsin(p) z= (t[sup:29hv4ru9]2[/sup:29hv4ru9]-s[sup:29hv4ru9]2[/sup:29hv4ru9])/2
I found the binding conditions to be equal to
-1 + s[sup:29hv4ru9]2[/sup:29hv4ru9]t[sup:29hv4ru9]2[/sup:29hv4ru9] < t[sup:29hv4ru9]2[/sup:29hv4ru9] - s[sup:29hv4ru9]2[/sup:29hv4ru9] < 1 - s[sup:29hv4ru9]2[/sup:29hv4ru9]t[sup:29hv4ru9]2[/sup:29hv4ru9]
I have the metric tensor and I know i just need to do a triple integral but how do I find the functions of s, t and p and how do I know the limits of integration?
any help would be greatly appreciated
thank you
-Felicity
-1+(x[sup:29hv4ru9]2[/sup:29hv4ru9]+y[sup:29hv4ru9]2[/sup:29hv4ru9]) < 2z < 1-(x[sup:29hv4ru9]2[/sup:29hv4ru9]+y[sup:29hv4ru9]2[/sup:29hv4ru9])
a) rewrite the binding condition in parabolic coordinates
b) using parabolic coordinates and the given metric tensor, find the volume of the solid
using x=stcos(p) y= stsin(p) z= (t[sup:29hv4ru9]2[/sup:29hv4ru9]-s[sup:29hv4ru9]2[/sup:29hv4ru9])/2
I found the binding conditions to be equal to
-1 + s[sup:29hv4ru9]2[/sup:29hv4ru9]t[sup:29hv4ru9]2[/sup:29hv4ru9] < t[sup:29hv4ru9]2[/sup:29hv4ru9] - s[sup:29hv4ru9]2[/sup:29hv4ru9] < 1 - s[sup:29hv4ru9]2[/sup:29hv4ru9]t[sup:29hv4ru9]2[/sup:29hv4ru9]
I have the metric tensor and I know i just need to do a triple integral but how do I find the functions of s, t and p and how do I know the limits of integration?
any help would be greatly appreciated
thank you
-Felicity