Begin2.DVI
Using the del operator ∇the divergence can be represented div v =∇·v =( ∂ ∂x eˆ 1 + ∂ ∂y ˆe 2 + ∂ ∂z ˆe 3 )·(v 1 ˆe 1 +v 2 ˆe ...
Example 7-11. Find the divergence of the vector field given by v =xyz eˆ 1 +yz^2 ˆe 2 +zxy^2 ˆe 3 Solution By definition div v ...
(xii) ∇(A·B) = (B·∇ )A+ (A·∇ )B+B×(∇× A) + A×(∇× B) (xiii) ∇·(∇u) = ∇^2 u= ∂^2 u ∂x^2 + ∂^2 u ∂y^2 + ∂^2 u ∂z^2 The op ...
Example 7-14. Let r =xˆe 1 +yˆe 2 +zˆe 3 denote the position vector to a general point (x, y, z ). Show that grad(^1 r ) = grad ...
Example 7-16. Let r =xˆe 1 +yˆe 2 +zˆe 3 denote the position vector to a general point (x, y, z )and let r=|r |. Find grad f(r ...
Example 7-18. Show that ∇× (∇× A) = ∇(∇·A)−∇^2 A Solution Calculate ∇× A using determinants to obtain ∇× A= ∣∣ ∣∣ ∣∣ ˆe 1 ˆ ...
in terms of the arc length parameter s, where sis measured from some fixed point on the curve. In general, the scalar field φ=φ( ...
the greatest rate of change of φ. The length of the gradient vector is |grad φ|and represents the magnitude of this greatest rat ...
is a vector tangent to the curve of intersection and perpendicular to both of the normal vectors ∇F and ∇G. A unit tangent vecto ...
angles specify the directions one should travel in order to achieve the maximum (or minimum) rate of change of the scalar φ. A p ...
the surface. If the surface is a closed surface, then one normal is called an inward normal and the other an outward normal. Max ...
f(x)> f (x 0 )) holds. The determination of relative maximum and minimum values of a differential function y=f(x)over an inte ...
The problem of determining the relative maximum and minimum values of a function of two variables is now considered. In the disc ...
of the scalar field z which represents the height of the curve.) To picture what the above equations are describing, let x=x 0 + ...
more tractable for analysis. Factor out the leading term and then complete the square on the first two terms to obtain d^2 z ds^ ...
Hence, if A> 0 , then A(AC −B^2 ) is negative and alternatively if A< 0 , then A(AC −B^2 )is positive. In either case, the ...
condition for f=f(x, y )to have an extremum value at a point (a, b)requires that the differential df = 0 or df =∂f ∂x dx +∂f ∂y ...
The equations (7.78) can be interpreted that when a member of the family of curves f(x, y ) = c=constant is tangent to the const ...
of stationary values of f with a constraint condition because one would have at a stationary value of F the conditions ∂F ∂x =∂f ...
This implies that grad fis normal to the curve of intersection since it is perpendicular to the tangent vector dx ̄ to the curve ...
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