Coordinate changes and vector integral theorems
| English | 中文 | Pinyin |
|---|---|---|
| divergence/daɪˈvɜːdʒəns/ | 散度 | sàn dù |
| conservative field/kənˈsɜːvətɪv fiːld/ | 保守场 | bǎo shǒu chǎng |
A decision before an answer
- A vector field can have zero curl away from the origin while its circulation around the origin is nonzero. The hole in the domain matters.
- Your goal: Transform double and triple integrals with their Jacobians.
Read the relationship
- Changing variables scales area or volume by the absolute Jacobian determinant. Polar coordinates use r dr dtheta; cylindrical coordinates use r dr dtheta dz; spherical coordinates with phi measured from the positive z-axis use rho² sin(phi) dρ dphi dtheta. State angle conventions and transform both the integrand and the region. A missing Jacobian changes a uniform density integral into the wrong physical quantity.
- Apply Green, Stokes and divergence theorems with correct orientation.
What is the polar-coordinate area element?
The absolute Jacobian of (r cos(theta),r sin(theta)) is r.
Use the defining rule
- Green's theorem equates positively oriented planar boundary circulation integral P dx+Q dy with the double integral of Q_x−P_y on the region. The usual hypotheses require first derivatives continuous on an open set containing the region. A hole needs its own negatively oriented inner boundary, or another valid treatment of the missing domain. The theorem cannot integrate across a field singularity.
- Check domain singularities before asserting path independence.
For F=(x,y,z), what is the outward flux through the unit sphere?
Divergence is 3 and unit-ball volume is 4π/3, so outward flux is 4π.
Check the conditions
- Stokes' theorem equates circulation on a surface boundary with the surface integral of curl F dot the oriented normal. The right-hand rule links boundary direction to the normal. The divergence theorem equates outward flux across a closed surface with the volume integral of div F. Circulation, flux, curl and divergence are distinct; a closed surface is required for the usual divergence theorem.
- Check domain singularities before asserting path independence.
For F=(x,y,z), divergence is 3. The outward flux through the sphere of radius 2 is therefore 3 times its volume, or 3·(4π·2³/3)=32π. This avoids a surface parameterisation. For the planar field (−y,x), Green's theorem gives counterclockwise unit-circle circulation as integral of 1−(−1)=2 over the disk, hence 2π. Reversing the orientation changes the circulation sign.
For F=(2x,3y,4z), the divergence is ____.
Differentiate matching coordinates and add 2+3+4.
Apply the task format
- A gradient field has path-independent line integrals, determined by endpoint potential differences. A continuously differentiable curl-free field on a simply connected open domain is conservative. Curl-free alone on a domain with a hole is insufficient. For F=(−y/(x²+y²),x/(x²+y²)), the origin is excluded; unit-circle circulation is 2π, although the curl is zero wherever the field is defined.
- Check domain singularities before asserting path independence.
Zero curl is not enough when the domain has a hole. Outward flux and counterclockwise circulation use different theorems and orientation rules.
Which answer fits this case?
Transform double and triple integrals with their Jacobians
A curl-free field on the punctured plane is always conservative.
The angular field in the lesson has zero curl away from zero but unit-circle circulation 2π, so it cannot be a global potential gradient there.
Keep the distinctions
- divergence 散度 — The sum of a vector field's coordinate-wise partial derivatives measuring local outward flow.
- conservative field 保守场 — A vector field equal to a scalar potential gradient with path-independent line integrals.
- Transform double and triple integrals with their Jacobians.
- Apply Green, Stokes and divergence theorems with correct orientation.
- Check domain singularities before asserting path independence.
Match each term with its precise meaning in this lesson.
Keep the distinctions stated in the teaching example.
Put this lesson’s reasoning or event sequence in order.
The order follows the stated process; check each stage before the next.