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Taylors series

- Before going into Gauss Theorem and Mass

conservation it will be useful to review Taylor

series. The concept of expanding a function about

a point will be used many times throughout this

course and in SO 414. - If a function converges about the point xo , we

can represent the function as a power series - Note The term means terms of

4th order and higher.

Taylors series

- The coefficients are

evaluated by examining the derivatives of f(x) at

xxo - where

Taylors series

- Substitution of the preceding coefficient values

into the power series expansion yields the Taylor

series of the function - Notice there are no formal restrictions on the

selection of the point xo besides the requirement

that the function must converge. - There are some practical limitations however such

as choosing xo near the center of the domain, so

that we dont have to consider an overwhelming

number of terms in the Taylor expansion.

Taylors series

- To ensure the series converges in a limited

number of terms, we will restrict the domain such

that - Then we can substitute in

the Taylor series where - The Taylor series is then expressed as
- The advantage of this form is we know the series

converges and we only need to consider a couple

terms

Taylors series

- Since it is safe to assume accuracy with only a

couple terms we will approximate the Taylor

Series as

Taylors series

- In rectangular coordinates, the multivariable

Taylor expansion is fairly straightforward. By

considering variations in only one variable at a

time, we are essentially looking at three

different 1-D cases of the previous equation. - If we consider variation in all three direction,

we obtain - With and

Gauss Theorem

- Now we utilize the Taylor series to derive a

theorem that will be of great value to us. - Gauss Theorem
- Given a fixed volume V which is bounded by the

area A and unit vector normal to the

surface element dA - For a uniformly continuous vector field,
- Gauss theorem states

Gauss Theorem - proof

- Consider an infinitesimal cube with sides dx,dy

and dz as shown where the center of the cube is

at position xo, yo and zo.

Gauss Theorem - proof

- Now examine the flow out of each face of the cube.

Gauss Theorem - proof

- Now use Taylor series expansions on all the

terms. - For example, the first two terms have Taylor

series expansions - Adding these two terms together we obtain

Gauss Theorem - proof

- Performing the same analysis on the y and z cube

faces we obtain - and

Gauss Theorem - proof

- Combining all of the above expressions we obtain

the result - Since the above result was performed on an

infinitesimal cube, we can sum up (or Riemann

integrate) all the cubes that make up an

arbitrary shape. - In doing such a summation we observe that the

vector contributions of adjacent faces are equal

and opposite to each other and cancel each other

out. Such cancellations will take place

throughout the entire interior so that we are

left only with the contributions through the

surface of our arbitrary shape.

Gauss Theorem - proof

- The result after taking the summation under the

limit of an infinite number of cubes whose volume

approaches zero is Gauss Theorem - A more common notation convention is to imply the

triple and double integrals over the volume and

surface area respectively

Exercise Archimedes Principle

- The net force due to hydrostatic pressure on a

submerged object is mathematically described as - Where is the outward normal to the

submerged object. - You may assume the medium is incompressible so
- rwater constant
- Use Gauss theorem to show that the above net

buoyant force of the surface of the object is

also equal to the weight of the displaced water

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