Jacobians and change of variables practice problems
The Jacobian ∂(x, y)/∂(u, v) of a change of variables, and the area scaling it measures. 10 problems with worked solutions; in 10 of them every equation is proved by a computer algebra system.
Find the Jacobian \( \displaystyle \dfrac{\partial(x, y)}{\partial(u, v)} \) of the transformation \( \displaystyle x = u v \), \( \displaystyle y = u - v \).
Find the Jacobian \( \displaystyle \dfrac{\partial(x, y)}{\partial(u, v)} \) of the transformation \( \displaystyle x = u \cos{\left(v \right)} \), \( \displaystyle y = u \sin{\left(v \right)} \).
Find the Jacobian \( \displaystyle \dfrac{\partial(x, y)}{\partial(u, v)} \) of the transformation \( \displaystyle x = u + v^{2} \), \( \displaystyle y = v \).
Find the Jacobian \( \displaystyle \dfrac{\partial(x, y)}{\partial(u, v)} \) of the transformation \( \displaystyle x = \frac{u}{v} \), \( \displaystyle y = v \).
Find the Jacobian \( \displaystyle \dfrac{\partial(x, y)}{\partial(u, v)} \) of the transformation \( \displaystyle x = - 2 u - v \), \( \displaystyle y = - 3 u + 3 v \).
Find the Jacobian \( \displaystyle \dfrac{\partial(x, y)}{\partial(u, v)} \) of the transformation \( \displaystyle x = e^{u} \cos{\left(v \right)} \), \( \displaystyle y = e^{u} \sin{\left(v \right)} \).
Find the Jacobian \( \displaystyle \dfrac{\partial(x, y)}{\partial(u, v)} \) of the transformation \( \displaystyle x = - 4 u - 2 v \), \( \displaystyle y = 2 u + 3 v \).
Find the Jacobian \( \displaystyle \dfrac{\partial(x, y)}{\partial(u, v)} \) of the transformation \( \displaystyle x = u - 4 v \), \( \displaystyle y = 3 u - v \).
Find the Jacobian \( \displaystyle \dfrac{\partial(x, y)}{\partial(u, v)} \) of the transformation \( \displaystyle x = u + 4 v \), \( \displaystyle y = - 3 u + 3 v \).
Find the Jacobian \( \displaystyle \dfrac{\partial(x, y)}{\partial(u, v)} \) of the transformation \( \displaystyle x = u^{2} - v^{2} \), \( \displaystyle y = 2 u v \).