Partial and Total Differentials

Excerpts

• Partial and Total Differentials

  Therefore the numerator of each of the fractions $\dfrac{p}{a}$, $\dfrac{p}{b}$, and $\dfrac{p}{c}$, must never be separated from its denominator, because the value of the former depends, in part, upon the latter; and one $p$ cannot be distinguished from another without its denominator.

• Partial and Total Differentials

  The last equation gives a striking illustration of the method of notation. Treated according to the common rules of algebra, it is $du = du + du$, which is absurd, but which appears rational when we recollect that the second $du$ arises from a change in $x$ only, the third from a change in $y$ only, and the first from a change in both.

• Partial and Total Differentials

  The symbol $\phi(x, y)$ must not be confounded with $\phi(xy)$. The former represents any function of $x$ and $y$; the latter a function in which $x$ and $y$ only enter so far as they are contained in their product.

• Partial and Total Differentials

  Neither are we allowed to say that $\dfrac{p}{a}$ divided by $\dfrac{p}{b}$ is $\dfrac{b}{a}$; for this supposes that $p$ means the same thing in both quantities.

• Partial and Total Differentials

  The *etc.* is the representative of an infinite series of terms, the aggregate of which diminishes continually with respect to $dp$, $dq$, etc., as the latter are diminished, and which, therefore, has no effect on the *limit* of the ratio of $d.z$ to any other quantity.

• Partial and Total Differentials

  in which, however, it must be remembered, that $du$ does not stand for the same thing in any two of the three equations: it is true that it always represents an increment of $u$, but as far as we have yet gone, we have used it indifferently, whether the increment of $u$ was the result of a change in $x$ only, or $y$ only, or both together.

Equations

• Partial and Total Differentials

  u = x^{2} y + 2xy^{3}

  Defines the function u as x squared times y plus twice x times y cubed, the example used throughout the chapter.

• Partial and Total Differentials

  du = (2xy + 2y^{3})\, dx + \etc.

  When only x varies, the increment of u is (2xy + 2y^3) dx plus terms that become negligible as dx diminishes.

• Partial and Total Differentials

  du = (x^{2} + 6xy^{2})\, dy + \etc.

  When only y varies, the increment of u is (x^2 + 6xy^2) dy plus terms negligible as dy diminishes.

• Partial and Total Differentials

  du = (2xy + 2y^{3})\, dx + (x^{2} + 6xy^{2})\, dy + \etc.

  When x and y both vary, the increment of u is the sum of the two partial contributions plus negligible terms.

• Partial and Total Differentials

  \dfrac{du}{dx}\, dx = (2xy + 2y^{3})\, dx

  The x-part of the increment of u is the partial differential coefficient of u with respect to x times dx, here equal to (2xy + 2y^3) dx.

• Partial and Total Differentials

  \frac{du}{dx} = 2xy + 2y^{3}

  The partial differential coefficient of u with respect to x equals 2xy + 2y^3 for this example.

• Partial and Total Differentials

  \dfrac{du}{dy}\, dy = (x^{2} + 6xy^{2})\, dy

  The y-part of the increment of u is the partial differential coefficient of u with respect to y times dy, here equal to (x^2 + 6xy^2) dy.

• Partial and Total Differentials

  \frac{du}{dy} = x^{2} + 6xy^{2}

  The partial differential coefficient of u with respect to y equals x^2 + 6xy^2 for this example.

• Partial and Total Differentials

  d.u = \frac{du}{dx}\, dx + \frac{du}{dy}\,dy

  The total differential of u, the increment from both suppositions at once, is the sum of its two partial contributions.

• Partial and Total Differentials

  \ux\, dx + \uy\, dy + \etc

  The increment of u when x and y both receive increments is the sum of the partial terms in dx and dy, with the remaining terms negligible in the limit.

• Partial and Total Differentials

  dz = \frac{dz}{dp}\, dp + \frac{dz}{dq}\, dq + \frac{dz}{dr}\, dr + \frac{dz}{ds}\, ds + \etc.

  For z a function of p, q, r and s, the total differential of z is the sum of the partial contributions from each variable, plus terms negligible in the limit.

Problems

No exercises in this chapter.