The application of Jacobian determinant in the transformation of integral coordinates Zou Zemin systematically discussed the important application of Jacobian determinant in the transformation of function integral coordinates. Because the relationship between (x, y) and (u, v) is not necessarily linear, only reversible linear relationship can ensure that straight lines can be mapped into straight lines, and generally only straight lines can be mapped into curves.

The most commonly used are polar coordinate substitution in double integral, spherical coordinate substitution in triple integral and cylindrical coordinate substitution, all of which are obtained by Jacobian determinant. In addition, the derivative of the function determined by the implicit function F(x, y, z)=0 at high numbers is also obtained by Jacobian determinant.

meaning

If the Jacobian determinant is not zero everywhere in a connected region, it is positive or negative everywhere (its symbol indicates whether the rotation directions of the U coordinate system and the X coordinate system are consistent). This way. Under the condition that the Jacobian determinant is not equal to zero, the continuously differentiable function set establishes a one-to-one correspondence between the starting point and the points near each pair of corresponding points U and X. ..