Differentiation of the determinant of the Jacobian

Question

I am working through A Mathematical Introduction to Fluid Mechanics and I have come to a statement on showing what I am guessing is a corollary to Jacobi's Formula https://en.wikipedia.org/wiki/Jacobi%27s_formula

I do not understand the step on the next page of this text where they use multilinearity of the determinant to apply the derivative in this way. It looks like that they use differentiation similar to a scalar, but I do not understand how or why they are adding these Jacobians.

https://books.google.com/books?id=HBXSBwAAQBAJ&pg=PA8&lpg=PA8&dq=The+determinant+J+can+be+differentiated+by+recalling+that+the+determinant+of+a+matrix+is+multilinear+in+the+columns&source=bl&ots=2nBkkqc4sZ&sig=ACfU3U3MY4WFxZdLCITJUyywE7B_qaKcoA&hl=en&sa=X&ved=2ahUKEwiK5_2Fq4rgAhVNPN8KHZRNAsQQ6AEwAHoECAQQAQ#v=onepage&q=The%20determinant%20J%20can%20be%20differentiated%20by%20recalling%20that%20the%20determinant%20of%20a%20matrix%20is%20multilinear%20in%20the%20columns&f=false

Answer 1

The Jacobian is used in coordinate transformations. Picture the xy plane parallel to a slightly "curved plane" (let's say u,v). Imagine lines (vectors) taking all the points from x,y and putting them on a new plane, u,v. When doing so, we need to scale things from one plane to the other. That's my simplified understanding and over generalization of it.

Determinants are used to calculate area or volume, also.

You are right about thinking of it as a scalar.

As far as the adding three determinants when taking the derivative of the Jacobian,

You can multiply a row or column of a determinant by a scalar, or you can divide a row or column from a determinant: the operation only includes one row or one column. This does not change the value of the determinant.

Sorry trying to type on my phone.