Begin2.DVI
∂F ∂x = ∂F 1 ∂x ˆe 1 +∂F^2 ∂x ˆe 2 +∂F^3 ∂x ˆe 3 ∂F ∂y = ∂F 1 ∂y ˆe 1 +∂F^2 ∂y ˆe 2 +∂F^3 ∂y ˆe 3 ∂F ∂z = ∂F 1 ∂z ˆe 1 +∂F^2 ...
Notation The position vector r =xˆe 1 +yˆe 2 +zˆe 3 is sometimes represented in matrix no- tation as a row vector r = (x, y, z ...
The divergence of a vector function F(x, y, z)is a scalar function defined by div F = ∂F 1 ∂x + ∂F 2 ∂y + ∂F 3 ∂z If one uses ...
so that if φrepresents either of the components F 1 or F 2 one can write (h·∇ )φ=h (^1) ∂x∂φ 1 +h (^2) ∂x∂φ 2 Observe that the ...
Differentiation of Composite Functions Let φ=φ(x, y, z)define a scalar field and consider a curve passing through the region whe ...
Higher derivatives can be calculated by using the product rule for differentiation together with the rule for differentiating a ...
Example 6-27. The acceleration of a particle is given by a = sin tˆe 1 + cos tˆe 2. If at time t= 0 the position and velocity o ...
Example 6-28. A particle in a force field F=F(x, y, z )having a position vector r =xˆe 1 +yˆe 2 +zˆe 3 moves according to New ...
where x, y, z define some parametric representation of the curve C. The element of arc length along the curve, when squared, is ...
s 0 < s 1 <... < sn, where corresponding to each value of the arc length parameter sithere is a position vector r (si) ...
∫ C F·dr = lim n→∞ ∑n i=1 F(x∗i, y ∗i, zi∗)· ∆ri ∆si ∆si = ∫ C ( F 1 dx ds +F 2 dy ds +F 3 dz ds ) ds, = ∫ C F·dr = ∫ C F ...
Note that each of the line integrals requires knowing the values of x,y and z along a given curve Cand these values must be subs ...
which is determined by the vector force field. Newton’s second law of motion is expressed F =ma =md (^2) r dt^2 =m dv dt. Th ...
Here F·ˆetis the tangential component of the force F along the given curve C. This form of the line integral is used if F =F ...
to be in the negative sense if the direction of integration is clockwise. The sense of integration is the same as that for angul ...
Using the property that line integrals may be broken up into integration along separate curves, one can write W= ∫B O F·dr = ∫ ...
then the above line integral can be written ∫ C ©F·dr = ∫ C (xˆe 1 +yeˆ 2 )·(dx ˆe 1 +dy ˆe 2 ) = ∫ C x dx +y dy = ∫ 2 π 0 (co ...
Solution: The work done is determined by evaluating the line integral ∫ CF·dr where F·dr = (x+z)dx + (y+z)dy + 2z dz On the ...
Exercises 6-1. For the vectors A= 3 ˆe 1 + 2 ˆe 2 +ˆe 3 and B= 6 ˆe 1 −ˆe 2 + 2 ˆe 3 calculate (a) A+B (b) 6A− 3 B (c) A ...
6-9. Consider the triangle defined by the three vertices (6 , 0 ,0),(0 , 6 ,0) and (0 , 0 ,12). Use vector methods to find the ...
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