Let the temperature T in a body be independent of z so that it is given by a scalar function:
T = 9x^2 + 4y^2 .
Identify the isotherms T(x,y) = const.
they are just ellipses parallel to the xy plane.
To identify the isotherms, we need to find the curves in the (x, y) plane that satisfy the equation T(x, y) = const.
In this case, the isotherms will be the curves that have a constant temperature value. Let's set T(x, y) equal to a constant, represented by C:
T(x, y) = 9x^2 + 4y^2 = C
To identify the specific curves in the (x, y) plane, we can manipulate the equation to represent y in terms of x:
4y^2 = C - 9x^2
y^2 = (C - 9x^2) / 4
y = ±√((C - 9x^2) / 4)
Now, we have the equation of the isotherms in the form y = f(x), where f(x) is given by ±√((C - 9x^2) / 4). Note that the plus and minus signs represent two curves on either side of the y-axis.
For different values of C, we obtain different isotherms. By selecting various values of C, we can generate a family of curves that represent the isotherms for the given temperature function.