座標変換とベクトル積分定理
導入済み| English | 意味 |
|---|---|
| divergence/daɪˈvɜːdʒəns/ | 発散 |
| conservative field/kənˈsɜːvətɪv fiːld/ | 保存場 |
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?
難易度 1/5 · 100 コイン + 今日初回完了時に小ボーナス付与
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?
難易度 1/5 · 100 コイン + 今日初回完了時に小ボーナス付与
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 ____.
難易度 1/5 · 100 コイン + 今日初回完了時に小ボーナス付与
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.
このケースに当てはまる答えはどれか？
ヤコビアンを用いた二重・三重積分の変換
A curl-free field on the punctured plane is always conservative.
難易度 1/5 · 100 コイン + 今日初回完了時に小ボーナス付与
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.
難易度 2/5 · 140 コイン + 今日初回完了時に小ボーナス付与
Keep the distinctions stated in the teaching example.
Put this lesson’s reasoning or event sequence in order.
難易度 3/5 · 200 コイン + 今日初回完了時に小ボーナス付与
The order follows the stated process; check each stage before the next.