Appendix A: Leibniz's Rule

Leibniz's Rule is to find the derivative with respect to t of a quantity

# ( t ) which is an integral of a function f of variables over a domain that is a function o f t . Geometrically, the domain of integration of variables can be a space that varies with time t. We derive Leibniz's rule for the integral of an arbitrary function f in one dimension, two dimensions, and three dimensions to illustrate its geometric meaning.

A.l One 1)imensional Intepral

dA(t) A(t + At) - A(1) = lim 1 dl A ~ + O A/ = A,+O lim [ " A ' ( x / ~1 , ( I + A I ) + A ) - f'("f(x,/)dx ( ( 1 1 1

From Taylor series expansion, db b(t + At) = b(t) +- A t + ... dl

We have, to the limit of At + 0

Therefore, the Leibniz's rule for 1 -D integral is written.

Let's now examine the geometric intcrpretation as show in Fig. A l , where A

(t) is the area obtained by the integration off (x, t) with respect to x, over the one - di~nensional domain (a (t), b (t)). Ilere, a and b are the boundary points of (a, b).

da . db . In tcrllxi or vector, the velocitie s at a and b arc v a = - I and V , = - I, rcspcctivc ly

dt dt From Figure A l , Let's define the unit vector outward normal to dolnain

bo~~ndaries as n. Then, for 1-D case, where the domain is given by (a, b),

n,=-i and nb=i, where i is the unit vector in x-direction. Then the 1,eibniz's r ~ ~ l e in vector notation becomes

-.-. I,\\\\\, ,,\,\\, , ., ... Arecr Clzcrnge jrom Time t to t + V t

T .f' - - - - I

A.2 Two Dimensional lntegral

Let's consider the 2-D case for the integral V(t) obtained by the integration o f f (x, y, t) over the domain A(t) that change with t.

where, A(t) is the area (domain) laying on the x-y plane with dA=dxdy and is enclosed by the boundary line-loop C(t) as iliustrated in Figure A.2. The change o f V(t) with respect to t, now, according to L,eibniz's rule, can be written as

( I ) -dA + 4 f d c d n dt 0 ' dt

where dA at the boundary C(t) is expressed as dA=dndc. Hence, (n,c) is the local coordinate with dc being tangential and dn being normal to C(t).

If v is the velocity on C(t), the velocity co~nponent of v outward normal to tin C(/) is v. =v* n=-, the outward normal velocity vector. I-lere, we have, dt

where d c = dc n is the outward normal vector of dc on C(t) The 1,eibniz' s rule in 2 - D then becomes :

J x Fig. A.2

A.3 Threc I)imensional Integral

Let's consider the 3-D case where the integral, K(t) is obtained by the intcgral o f f (x, y, z, t) over the volurne domain V(/) that change with t,

Hence, V(t) is the volume (domain) laying on the x-y-z space with dV-dxdydz and is enclosed by the surl'ace boundary S(t) as illustrated in 1;igure A.3. The change ol'K(t) with respect to t , now, according to 1,eibniz3s rulc, can bc writtcn as

where dV = ~Inds wit11 CIJ. being the area increament on S and (in being the increament in the direction outward normal to S

dV d n NO^ lhul, - - - -ds = v , , d s = v n d s = v U'S

dl d/

dn where - = v , , = v a n

' dl d s = n d s In1 = 1

Summary

"f' can be any quantity either a scalar, vector or tensor. Iffis any tensor, e.g. density, energy of ''f(x, y, z, and t)", which is fi~nction of x, y, z and t , then it is true that:

K f i ) = l l ~ , ~ , , , f f x , . ~ , z , ~ ) ~ / x d ~ d z