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.
To identify the isotherms of the temperature function T(x, y) = 9x^2 + 4y^2, we need to find the curves in the (x, y) plane where the temperature is constant.
To do this, we can set the temperature function equal to a constant, say C, and solve for either x or y. Let's solve for y since it makes the equation simpler:
9x^2 + 4y^2 = C
Divide both sides by the constant C:
(9x^2)/C + (4y^2)/C = 1
Now, we can rewrite this equation in terms of y:
y^2 = (C - (9x^2))/4
To find the isotherms, we can plot these curves by choosing different values of C and finding the corresponding values of y for a given x.
For example, if we take C = 4, the equation becomes:
y^2 = (4 - 9x^2)/4
Simplifying further:
y^2 = 1 - (9/4)x^2
Taking the square root of both sides:
y = ±√(1 - (9/4)x^2)
By plotting these curves for different values of C, we can identify the isotherms of the temperature function T(x, y) = 9x^2 + 4y^2. The isotherms are the curves where the temperature is constant.