Chapter 6: Vector Analysis
Q15P
Suppose the density varies from point to point as well as with time, that is, . If we follow the fluid along a streamline, then are function of such that the fluid velocity is
Show that then . Combine this equation with to get
(Physically, is the rate of change of density with time as we follow the fluid along a streamline; is the corresponding rate at a fixed point.) For a steady state (that is, time-independent), , but is not necessarily zero. For an incompressible fluid, . Show that then role="math" localid="1657336080397" . (Note that incompressible does not necessarily mean constant density since does not imply either time or space independence of ; consider, for example, a flow of watermixed with blobs of oil.)
Q15P
In the figure is a unit vector in the direction of an incident ray of light, and and are unit vectors in the directions of the reflected and refracted rays. If is a unit vector normal to the surface , the laws of optics say that and , where and are constants(indices of refraction). Write these laws in vector form (using dot or cross products).
Q15P
, where C is the curve of intersection of the surfaces whose equations are .
Q16P
Question:What is wrong with the following “proof” that there are no magnetic fields? By electromagnetic theory,∇· B = 0, and B =∇×A. (The error is not in these equations.) Using them, we find
Since, A is conservative, or A =∇ψ. Then ,so B = 0.
Q16P
In the discussion of Figure 3.8, we found for the angular momentum, the formula .Use (3.9) to expand this triple product. If is perpendicular to , show that you obtain the elementary formula, angular momentum .
Q16P
Given and
(a) Are these forces conservative? Find the potential corresponding to any conservative force.
(b) For any nonconservative force, find the work done if it acts on an object moving from (-1,-1)to (1,1)along each of the paths shown.
Q16P
The following equations are variously known as Green’s first and second identities or formulas or theorems. Derive them, as indicated, from the divergence theorem.
To prove this, let in the divergence theorem.
To prove this, copy Theorem above as is and also with and interchanged; then subtract the two equations.
Q16P
Given
(a) Which F , if either, is conservative?
(b) If one of the given ’s is conservative, find a function Wso that
(c) If one of the F’s is non conservative, use it to evaluate along the straight line from
(d) Do part (c) by applying Green’s theorem to the triangle with vertices .
Q16P
Show by the Lagrange multiplier method that the maximum value of .That is, maximize given by (6.3) subject to the condition . You should get two values () for the Lagrange multiplier λ, and two values (maximum and minimum) forwhich is the maximum and which is the minimum?
Q17MP
Find the value of along the circle from (1,1) to (1,−1) if F= (2x−3y)i−(3x−2y)j.