Multivariable calculus and vector analysis
| English | 中文 | Pinyin |
|---|---|---|
| gradient/ˈɡreɪdɪənt/ | 梯度 | tī dù |
| Jacobian/dʒæˈkəʊbɪən/ | 雅可比行列式 | yǎ kě bǐ háng liè shì |
A decision before an answer
- A hillside has many slopes at one point. The direction of travel determines which slope you experience.
- Your goal: Compute partial derivatives, gradients and directional derivatives.
Read the relationship
- A partial derivative changes one coordinate while fixing the others. For f=x²+3xy, f_x=2x+3y and f_y=3x. The gradient collects these derivatives; if f is differentiable, the directional derivative along a unit vector v is grad f·v. Normalise the direction before taking this dot product. A direction vector of length two would double the answer if used without normalisation.
- Use multiple integrals and coordinate changes.
Partial derivative with respect to x of x²y is:
Treat y as constant.
Use the defining rule
- Differentiability means a valid linear approximation, not merely the existence of some partial derivatives at one point. Continuous first partials in a neighbourhood are a sufficient condition. For a composition f(x(t),y(t)), the chain rule gives f_x x′+f_y y′. Second derivatives can introduce both direct and mixed terms; a zero mixed partial alone places no restriction on the pure second partials.
- Use derivative conditions to distinguish planes from curved surfaces.
The polar area element is:
The Jacobian of the polar transformation is r.
Check the conditions
- A multiple integral sums contributions over a specified region. Describe the region before choosing iterated limits; nonrectangular bounds may change when the order changes. In polar coordinates area is r dr dtheta, not just dr dtheta. A general coordinate change uses the absolute Jacobian determinant. Sign belongs to oriented vector quantities, while area and volume scaling use a nonnegative factor.
- Use derivative conditions to distinguish planes from curved surfaces.
For f(x,y)=x²+3y², gradient f=(2x,6y). At (1,1) this is (2,6). In direction (3,4), the unit vector is (3/5,4/5), giving directional derivative 2·3/5+6·4/5=6.
The magnitude of vector (3,4) is ____.
Use sqrt(3²+4²)=5.
Apply the task format
- If both first partials of a globally defined function on R² are constant, f_x=a and f_y=b, integrating successively gives f=ax+by+c, a plane. This cannot be inferred from parallel straight level sets alone: e^x has vertical parallel level lines but a curved graph. Nor do f_xy=f_yx=0 force a plane: x²+y² is a counterexample. Distinguish first-derivative constancy from absent mixed dependence and from the shape of selected level sets.
- Use derivative conditions to distinguish planes from curved surfaces.
A non-unit direction vector gives a scaled directional rate, not the derivative per unit distance.
Which answer fits this case?
Compute partial derivatives, gradients and directional derivatives
Orientation never affects a line integral around a boundary.
Reversing orientation changes a directed line integral’s sign.
Keep the distinctions
- gradient 梯度 — Vector of partial derivatives.
- Jacobian 雅可比行列式 — The local area or volume scaling in a coordinate change.
- Compute partial derivatives, gradients and directional derivatives.
- Use multiple integrals and coordinate changes.
- Use derivative conditions to distinguish planes from curved surfaces.
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.