--%>
Assignment Help - homework help

  Differential Calculus, Physics tutorial

Partial Derivatives:

The partial derivative of the function of numerous variables is its derivative with respect to one of those variable with others held constant.

x = x(z,y)........................................Eq.1

From equation 1, x a dependent variable is the function of two independent variables z and y . Partial derivative of x with respect to y with z held constant is (∂x/∂y)z

For example, if

x = zy2........................................Eq.2

Then, partial derivative of x with respect to y with z held constant is

(∂x/∂y)z = 2zy........................................Eq.3

Likewise, partial derivative of x with respect to z with y held constant is

(∂x/∂z)y = y2........................................Eq.4

Exact Differential:

Assume that there exists the relation among three coordinates x, y, and z in such a way that x is a function of y and z (i.e. x(z, y) ); therefore

f(x, y, z) = 0........................................Eq.5

Exact differential of x (dx) is

dx(∂x/∂y)z dy + (∂x/∂z)ydz........................................Eq.6

Usually for any three variables x , y , and z we have relation of form

dx = M(y,z)dy + N(y,z)dz........................................Eq.7

If differential dx is exact, then

(∂M/∂z)y = (∂N/∂y)z........................................Eq.8

Implicit Differential:

Consider the equation of form

xy = x2y2........................................Eq.9

One can differentiate two sides of equation 9 using equation 6 (i.e differentiating both the left and right hands side with respect to x while y is held constant and with respect to y while x is held constant).

(∂(xy)/∂x)ydx + (∂(xy)/∂y)xdy = (∂(x2y2)/∂x)ydx + (∂(x2y2)/∂y)xdy........................................Eq.10

Equation 10 gives

ydx + xdy = 2xy2dx + 2x2ydy........................................Eq.11

Collecting like term and then factorize to have

dy/dx = (2xy2 - y)/(x - 2x2y) ........................................Eq.12

Consider equation 9 as f = xy - x2 y2 (i.e. moving expression in right side of equation 9 to left side and then equate result to f ). Then

dy/dx = -(∂f/∂x)/(∂f/∂y) ........................................Eq.13

Product of Three Partial Derivatives:

Assume that there exists the relation among three coordinates x, y, and z; therefore

f(x, y, z) = 0........................................Eq.14

Then x can be imagined as the function of y and z

dx = (∂x/∂y)zdy + (∂x/∂z)ydz........................................Eq.15

Also y can be imagined as the function of x and z, and

dy = (∂y/∂x)zdx + (∂y/∂z)xdz........................................Eq.16

Insert equation 16 in 15

dx = (∂x/∂y)z[(∂y/∂x)zdx + (∂y/∂z)xdz] + (∂x/∂z)ydz

Rearrange to get:

dx = (∂x/∂y)z(∂y/∂x)zdx + [(∂x/∂y)z(∂y/∂z)x + (∂x/∂z)y]dz........................................Eq.17

If dz = 0 dx ≠ 0 it follows that

(∂x/∂y)z(∂y/∂x)z = 1

(∂x/∂y)z = 1/((∂y/∂x)z)........................................Eq.18

In eq.17 if dx = 0 and dz ≠ 0, it follows that:

(∂x/∂y)z(∂y/∂z)x + (∂x/∂z)y = 0

Move (∂x/∂z)y to other side of equation to get

(∂x/∂y)z(∂y/∂z)x = -(∂x/∂z)y........................................Eq.19

Then divide both sides of equation 19 by (∂z/∂x)y

(∂x/∂y)z(∂y/∂z)x(∂z/∂x)y = -1........................................Eq.20

This is known as minus-one product rule.

Chain Rule of Partial Derivatives:

Another helpful relation is known as chain rule of partial derivatives. Assume T is function of V and P, and that each of V and P is the function of Z, then

(∂T/∂V)P = (∂T/∂Z)P(∂Z/∂V)P........................................Eq.21

Equation 21 is chain rule of partial derivative. The following can as well be written:

(∂S/∂P)T = (∂S/∂V)T(∂V/∂P)T........................................Eq.22(a)

(∂U/∂V)P = (∂U/∂T)P(∂T/∂V)P........................................Eq.22(b)

Equation 21 and 22 are known as chain rule of partial derivatives

Second Derivatives or Second Order Derivatives:

Let f (x, y) be function with continuous order derivatives, then we can compute first derivatives to be (∂f/∂x)z and (∂f/∂z)x. One can further compute second derivatives ∂2f/∂x2, ∂2f/∂z2 ∂2f/∂x∂y, and ∂2f/∂y∂x. Take note of these two second order derivates i.e. ∂2f/∂x∂y and ∂2f/∂y∂x, they are known as mixed second derivatives. It can be shown that mixed second derivatives are equal, i.e. it doesn't matter order will perform differentiation.

∂2f/∂x∂y = ∂2f/∂y∂x

Functions of More than Two Variables:

Assume that f (x, y, z), derivative of f with respect to one of the variables with other two constant (e.g. derivative of f x with y and z constant) can be written as:

(∂f/∂x)yz, (∂f/∂y)xz, and (∂f/∂z)xy

Tutorsglobe: A way to secure high grade in your curriculum (Online Tutoring)

Expand your confidence, grow study skills and improve your grades.

Since 2009, Tutorsglobe has proactively helped millions of students to get better grades in school, college or university and score well in competitive tests with live, one-on-one online tutoring.

Using an advanced developed tutoring system providing little or no wait time, the students are connected on-demand with a tutor at www.tutorsglobe.com. Students work one-on-one, in real-time with a tutor, communicating and studying using a virtual whiteboard technology.  Scientific and mathematical notation, symbols, geometric figures, graphing and freehand drawing can be rendered quickly and easily in the advanced whiteboard.

Free to know our price and packages for online physics tutoring. Chat with us or submit request at [email protected]