Let a parametrisation of some set be $$\varphi(x,\phi)=(x,(1-\sin x)\cos(\phi),(1-\sin x)\sin\phi)$$. This is an object I am interested in: $$\frac{d(\phi_2,\phi_3)}{d(x,\phi)}$$ What does an expression like this mean? It is the first time I have encountered it. It came from parametrising $dx_2\wedge dx_3$.
Any explanation appreciated, thanks I am new to this topic.
It denotes the Jacobian, that is the determinant of the Jacobi matrix: $$ \det \begin{bmatrix}\dfrac{\partial \phi_2}{\partial x}&\dfrac{\partial \phi_2}{\partial \phi}\\\dfrac{\partial \phi_3}{\partial x}&\dfrac{\partial \phi_3}{\partial \phi}\end{bmatrix} = \det \begin{bmatrix}\dfrac{\partial \big((1-\sin x)\cos\phi\big)}{\partial x}&\dfrac{\partial \big((1-\sin x)\cos\phi\big)}{\partial \phi}\\\dfrac{\partial \big((1-\sin x)\sin\phi\big)}{\partial x}&\dfrac{\partial \big((1-\sin x)\sin\phi\big)}{\partial \phi}\end{bmatrix}$$