Understanding Differentials: Deriving dε = F dot dr

[Syrus]

Homework Statement

The proof begins: Suppose that F is conservative. Then a scalar field ε(r) can be defined as the line integral of F from the origin to the point r. So ∫F dot dr = ε(r), where the limits of integration are from 0 to r.

The next step, however, eludes me: From the definition of an integral, it then follows that an infinitesimal change in ε is given by dε = F dot dr.

Homework Equations

The Attempt at a Solution

Usually total differentials are related to partial derivatives, tangent planes, and Taylor expansions. I'm failing to fill in the intermediate steps in deriving dε = F dot dr from ∫F dot dr = ε(r) using the "definition of integral". Any insight?

 

[Syrus]

Upon second thought:d/dr [∫F dot dr] = d/dr [∫F dot ndr] , where n is a unit vector in the direction of dr. So,

d/dr [∫F dot ndr] = d/dr [ε(r)]. Thus,

F dot n = d/dr [ε(r)]

and so, multiplying by dr:

dε(r) = F dot ndr = F dot dr

Is this valid?

 

[gabbagabbahey]

Upon second thought:d/dr [∫F dot dr] = d/dr [∫F dot ndr] , where n is a unit vector in the direction of dr. So,

d/dr [∫F dot ndr] = d/dr [ε(r)]. Thus,

F dot n = d/dr [ε(r)]

and so, multiplying by dr:

dε(r) = F dot ndr = F dot dr

Is this valid?

Close, but $\mathbf{r}$ is a vector, and so an infinitesimal change in $\epsilon ( \mathbf{r} )$ is really

$$d\epsilon = \frac{ \partial \epsilon}{ \partial x} dx + \frac{ \partial \epsilon}{ \partial y} dy + \frac{ \partial \epsilon}{ \partial z} dz = \mathbf{ \nabla } \epsilon \cdot d\mathbf{r}$$

 

[Syrus]

Hi gabbagabbahey! I understand that what you posted is the equation for the total differential, I am just struggling to understand how from ∫F dot dr = ε(r) (where the limits of integration are from 0 to r) one deduces the result- in particular, using the "definition of integral".

That is, how can we derive dε = F dot dr from what we have above?

 

[gabbagabbahey]

Hi gabbagabbahey! I understand that what you posted is the equation for the total differential, I am just struggling to understand how from ∫F dot dr = ε(r) (where the limits of integration are from 0 to r) one deduces the result- in particular, using the "definition of integral".

Well, to me, the statement "From the definition of an integral" means using the fundamental theorem of calculus (FTC). For a simple one-dimensional integral, FTC tells you that if $F(b)-F(a) = \int_a^b f(x) dx$, then $F'(x)=f(x)$ (or $dF = f(x)dx$).

For line integrals, this generalizes to the statement that if $F(\mathbf{b}) - F(\mathbf{a}) = \int_{ \mathbf{a} }^{ \mathbf{b} } \mathbf{f} ( \mathbf{r} ) \cdot d \mathbf{r}$ regardless of which path you choose for the integration, then $\mathbf{\nabla} F = \mathbf{f} ( \mathbf{r} )$ (or, equivalently $dF=\mathbf{f} ( \mathbf{r} ) \cdot d \mathbf{r}$)