Description
Excerpt from Vector Calculus
Analysis have been adopted from ''the simplest physical ideas which immediately identify them. Thus the idea of divergence is taken directly from Hydrodynamics, and keeping before our minds the picture of ?uid leaving (or entering) a small closed space, we define the divergence of a vector function at a point as the limit of the ratio, if one exists, of the surface-integral of the function over a small closed space surrounding the point to the volume enclosed by the surface, a unique limit being supposed to be reached by the closed surface shrinking up'' to a point in any manner.
Again, it is found that some vector fields can be specified completely by the gradient of a scalar function, so that the line integral T of the vector function along any closed curve in (simply connected) space would Vanish. Thus the work done is nil along any closed path in a conservative field of force. But in case the vector function cannot be so specified, an expression of this negative quality of the function at a point is naturally sought in its now non-evanescent line integral along a small closed (plane) path surrounding the point. The ratio of this line integral to the area enclosed by our path generally approa ches a limit as the path shrinks up to a point, independently of its original form and of the manner of its shrinking, but dep ending ou the orientation of its plane. The limit moreover has usually a maximum value, subject to the Variation of this orientation, and a vector of magnitude equal to this maximum value and drawn perpendicular to that aspect of the plane which gives us the maximum value is called the curl of the original vector function.
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This book is a reproduction of an important historical work. Forgotten Books uses state-of-the-art technology to digitally reconstruct the work, preserving the original format whilst repairing imperfections present in the aged copy. In rare cases, an imperfection in the original, such as a blemish or missing page, may be replicated in our edition. We do, however, repair the vast majority of imperfections successfully; any imperfections that remain are intentionally left to preserve the state of such historical works.
Analysis have been adopted from ''the simplest physical ideas which immediately identify them. Thus the idea of divergence is taken directly from Hydrodynamics, and keeping before our minds the picture of ?uid leaving (or entering) a small closed space, we define the divergence of a vector function at a point as the limit of the ratio, if one exists, of the surface-integral of the function over a small closed space surrounding the point to the volume enclosed by the surface, a unique limit being supposed to be reached by the closed surface shrinking up'' to a point in any manner.
Again, it is found that some vector fields can be specified completely by the gradient of a scalar function, so that the line integral T of the vector function along any closed curve in (simply connected) space would Vanish. Thus the work done is nil along any closed path in a conservative field of force. But in case the vector function cannot be so specified, an expression of this negative quality of the function at a point is naturally sought in its now non-evanescent line integral along a small closed (plane) path surrounding the point. The ratio of this line integral to the area enclosed by our path generally approa ches a limit as the path shrinks up to a point, independently of its original form and of the manner of its shrinking, but dep ending ou the orientation of its plane. The limit moreover has usually a maximum value, subject to the Variation of this orientation, and a vector of magnitude equal to this maximum value and drawn perpendicular to that aspect of the plane which gives us the maximum value is called the curl of the original vector function.
About the Publisher
Forgotten Books publishes hundreds of thousands of rare and classic books. Find more at www.forgottenbooks.com
This book is a reproduction of an important historical work. Forgotten Books uses state-of-the-art technology to digitally reconstruct the work, preserving the original format whilst repairing imperfections present in the aged copy. In rare cases, an imperfection in the original, such as a blemish or missing page, may be replicated in our edition. We do, however, repair the vast majority of imperfections successfully; any imperfections that remain are intentionally left to preserve the state of such historical works.
Details
Publisher - Forgotten Books
Language - English
Hardback
Contributors
Author
Durgaprasanna Bhattacharyya
Published Date -
ISBN - 9780428497019
Dimensions - 22.9 x 15.2 x 0.5 cm
Page Count - 98
Paperback
Contributors
Author
Durgaprasanna Bhattacharyya
Published Date -
ISBN - 9781440080388
Dimensions - 22.9 x 15.2 x 0.6 cm
Page Count - 100
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