Reading about “The vector calculus bridge project” (Tevian Dray and Corinne Manogue at Oregon State).
Their takeaways:
* key calculus idea: the differential (not limits)
* key derivative idea: rates of change (not slopes)
* key integral idea: total amounts (not areas/volumes)
* key curves/surfaces idea: “use what you know”? (not parameterization)
* key function idea: “data attached to the domain” (not graphs)
Differences in interpretation between mathematicians and physicists example 1:
If T(x,y) = k(x^2+y^2) then is T(r,\theta) = kr^2 or T(r,\theta) = k(r^2+\theta^2)? The first option is thinking about the meaning of the function in the world; the second is thinking about a function of two variables as an input-output relationship.
example 2:
physicists will use $$\hat{\theta}$$ to describe the direction of a magnetic field that points outwards from a wire. But this notation is missing from many math classes.
Reading Dray and Manogue, “Using differentials to bridge the vector calculus gap”, The College Mathematics Journal (2003).
Contrast the treatment of surface and line integrals in Stewart (math) and Griffiths (physics – see his electrodynamics book for a summary of calculus in 60 pages). They compare flux integrals over a sphere.
- (math) compute the normal vector via a parameterization of the surface in terms of two coordinates. Use the cross product.
- (physics) reason out the direction of the unit normal vector and the surface element size. Find the dot product between the vector field and the unit normal vector, then integrate.
The math version was general and did not involve geometric reasoning. The physics version used prior knowledge of the sphere.
Vector differentials: They recommend using “adapted basis vectors such as” $$\hat{r}, \hat{\theta}, \hat{\phi}$$ in addition to $$\hat{i}, \hat{j}, \hat{k}$$. To approach a paraboloid, they suggest using $$d\vec{r}$$ in cylindrical coordinates.
Reading Dray and Manogue, “Putting differentials back into calculus”, The College Mathematics Journal (2010).
“The differentials of Leigniz… capture the essence of calculus”. Think of d as “a little bit of” and an integral symbol as “a long S, and may be called… ‘the sum of'”.
Differentials are used in u-substitutions in integrals.
d(uv) = v du + u dv. This could lead to an implicit derivative (divide by du) or to relating rates (dt instead). “It is a statement about the relative rates”.
“The use of differentials turns the chain rule, implicit differentiation, and related rates… into something each rather than hard”. See Thompson’s 100 year-old text for differentials in the context of a textbook.