物理子衿
For any complex number 
, let 
, 
, and 
be its conjugate, modulus, and real part respectively. Show that
and 
.
Hence, or otherwise, show that for any complex numbers 
and 
,
.
Roughwork.
In the field 
of complex numbers, 
is called the real part of and 
the imaginary part, i.e.,
,
its trigonometric form being 
with 
the modulus and 
the argument, and its exponential form, 
.
This problem is not to be attempted.
Reading Comprehension
(Excerpted passages from Ch. II Sec. I by S. E. Coleman in A Text-Book of Physics, 1911.)
Matter exists in three physical states or conditions, called the solid, the liquid, and the gaseous states. Much of physics depends upon the characteristic properties which distinguish the states of matter from one another. Thus we have the mechanics of solids, the mechanics of liquids, and the mechanics of gases. These properties, therefore, require some attention at the outset.
Liquids are distinguished from solids by the fact that they tend to flow, and hence must be contained in vessels. Every solid, on the contrary, has a shape of its own, which it tends to preserve. Some solids, e.g. stone and iron, offer great resistance to a change of shape; others, such as wet clay and putty, can readily be molded into any form. But even the small amount of resistance offered by soft solids distinguishes them from liquids.
Many of the physical properties of gases may be learned from a study of the air, which is a mixture of several gases, principally nitrogen and oxygen. Although the air is everywhere about us, we are ordinarily unconscious of its existence unless it is in motion. When it is in motion, we recognize it as a current of air, a breeze, or a wind. We commonly call a vessel “empty” when it is full of air; and seldom stop to think that when the so-called empty vessel is being filled with a liquid or a solid, the air in it is being pushed out.
It will help toward clear thinking on this point to push an inverted tumbler into a vessel of water. The water does not rise to fill the tumbler, being prevented from doing so by the confined air; but when the tumbler is slowly inclined, the air escapes in a succession of bubbles, and the water enters at the same time to take its place.
This simple experiment shows that a body of air confined in any space tends to keep other bodies out of that space; and the same is true of all gases. But we know that, after a bicycle or an automobile tire is fully inflated, much air must still be pumped in to make it hard. Now air can be forced into the fully inflated tire only by compressing the air already in it into a smaller space; and experience teaches that the compression of the confined air can be carried as far as the strength of the tire or of the operator will permit. The great compressibility of air can be shown simply by pushing in the piston of a bicycle pump or other compression pump, while the outlet is closed with the finger. A vigorous push will compress the air perhaps to one half or even to one third of its original volume. When the piston is released, the air expands and drives it back.
All gases are highly compressible and expansible, like air. When any quantity of gas, however small, is admitted into an otherwise empty space it instantly expands so as to fill the space completely.
If the above experiment is repeated with the compression pump filled with water, it will be found that the water is as unyielding as a board, for the piston cannot be pushed in at all.
All liquids and most solids are only very slightly compressible. Even under very great pressure their change of volume is commonly so slight as to escape notice; and, for all practical purposes, they are regarded as incompressible. Hence great compressibility and expansibility are distinguishing properties of the gaseous state.