Mathematics of Physics and Engineering
126 Functions of Several Variables If the partial derivatives fxy and fyx exist and are continuous in some neighborhood of the ...
Functions, Sets, Gradient 127 / are surfaces. This observation leads to several possible ways to define curves and surfaces. Con ...
128 Functions of Several Variables (c) Let r = r(u,v) be a parametric representation of the surface. As- sume that the partial d ...
Functions, Sets, Gradient 129 we conclude that the vector Ax + 12 j — k defines the plane tangent to the surface at the point PQ ...
130 Functions of Several Variables || V/(Pfc)|| < 10 m for some positive integer m). 3.1.2 Integration and Differentiation Le ...
Integration and Differentiation 131 is, by definition, lG= w)L fdm (3 Li4) The idea of the Riemann integral was introduc ...
132 Functions of Several Variables We say that F is a potential vector field with potential / if F = V/ for some scalar field f. ...
Integration and Differentiation 133 THE AREA INTEGRAL. We now take the set G in (3.1.13) to be a subset of the plane K^2. The se ...
134 Functions of Several Variables the surface area element of S. EXERCISE 3.1.15. B Justify (3.1.19). Hint. Verify that ||r„ x ...
Integration and Differentiation 135 is the mass of fluid passing across the surface area element da per unit time. On page 154 w ...
136 Functions of Several Variables of finitely many simple closed piece-wise smooth curves. We say that the orientations of S an ...
Integration and Differentiation 137 divergence of F at the point P is the number div F(P) so that lim m(G)-»o m(G) div F(P) = li ...
138 Functions of Several Variables and, for a smooth simple closed planar curveC, the circulation ^ = §v-dr c measures the amoun ...
Integration and Differentiation 139 differential operator and the symbolic vector representations of the divergence and curl: di ...
140 Functions of Several Variables then V^2 F = V^2 Fi z+V^2 F 2 j+V^2 F 3 k. (cf Verify that V^2 (fg) = gV^2 f+ Given a referen ...
Curvilinear Coordinate Systems 141 vectors u> and ro, the vector w x r 0 is the velocity of the point with po- sition vector ...
142 Functions of Several Variables where the correspondence between the X and Q-coordinates is not invert- ible: the polar angle ...
Curvilinear Coordinate Systems 143 (ci,C2 + t,C3), t€R; The qz coordinate curve with q\= c\, q 2 = c 2 , is the collection of p ...
144 Functions of Several Variables then, using (3.1.36) and dropping (dt)^2 , we find that the line element in Q coordinates is ...
Curvilinear Coordinate Systems 145 the usual way; for example, 9f = nm /(gl+t,g2,g3)-/(gl,g2,g3) dqi t-»o t Unlike cartesian coo ...
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