- Calculus, Volume 2: Multi-Variable Calculus and Linear Algebra with Applications to Differential Equations and Probability
- Tom M. Apostol
- Second Edition
- 1991
- 978-1-119-49676-2
2.8 Exercises
In each of Exercises 3 through 12, a function $T: V_2 \rightarrow V_2$ is defined by the formula given for $T(x, y),$ where $(x, y)$ is an arbitrary point in $V_2.$ In each case, determine whether $T$ is one-to-one on $V_2.$ If it is, describe its range $T(V_2);$ for each point $(u, v)$ in $T(V_2),$ let $(x, y) = T^{-1}(u, v)$ and give formulas for determining $x$ and $y$ in terms of $u$ and $v.$
8.$\quad$ $T(x, y) = (e^x, e^y).$
Solution. $\quad$ Let $A = (x_0, y_0)$ and $B = (x_1, y_1)$ be points in $V_2$ with transformations $T(A) = (e^{x_0}, e^{y_0})$ and $T(B) = (e^{x_1}, e^{y_1}).$ By definition of the exponential function (Volume 1, Section 6.12), we know that for every real $x$ there is one and only one $y$ such that $\log y = x.$ Thus, if $e^{x_0} = e^{x_1}$ and $e^{y_0} = e^{y_1},$ then $x_0 = x_1$ and $y_0 = y_1.$ As a result, $T(A) = T(B)$ implies $A = B,$ which means $T$ is one-to-one on $V_2.$
$\quad$ Following the definition of the exponential function, the range $T(V_2)$ is the set of $(u, v)$ in $V_2$ where $u, v \gt 0.$ For each point $(u, v)$ in $T(V_2)$ we have $x = \log u$ and $y = \log v. \quad \blacksquare$