Calculus Volume 3 · 免费预览
多重积分中的换元
换元定理用变换后的形式替换被积函数和积分微元，绝对雅可比行列式对面积或体积加以修正，从而把极坐标、柱坐标和球坐标统一为特殊情形。
1. The two-variable theorem
The two-variable theorem
2. Computing Jacobians
Computing Jacobians
3. Three variables and the theorem's scope
Three variables and the theorem's scope
4. Choosing the transformation
Choosing the transformation
5. Change of variables formula
The double integral's change of variables theorem introduces which factor?
6. Polar Jacobian computation
Computing the Jacobian of x = r cos theta, y = r sin theta directly gives which value?
7. Inverse Jacobian
The Jacobian of the inverse transformation is which expression?
8. Three-variable Jacobian
The spherical coordinates' three-variable Jacobian equals which expression?
9. Why absolute value
The change-of-variables formula uses the absolute value of the Jacobian rather than the signed value because of what?
10. Jacobian sign handling
The change-of-variables formula uses the absolute value of the Jacobian. As a sign, what is taken? Answer with one word.
11. Jacobian determinant size two variable
The two-variable Jacobian is a determinant of size two by which number? Answer with a number.
12. Transformation regions
In the change of variables, the original region R corresponds to S in the new coordinates. As a single letter, what is S? Answer with an uppercase letter.
13. Linear map uniform scaling
For a linear change of variables, the Jacobian is constant, and the integral over a region scales by that constant times what?
14. Linear map scaling
A linear transformation scales all areas by the absolute Jacobian's constant. As a single word, what is it? Answer in lowercase.