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Appendices A. 4. 4 are demonstrated. As an application, these laws are used to show graphically and mathematically how vectors may be bisected. ) B. 5. The divergence is developed again in cylindrical coordinates as a first-level generalization towards curvilinear coordinates taking into account that the azimuthal φ coordinate is the sole curvilinear coordinate in the cylindrical system. Coordinate conversions and differentials, metric coefficients, differential elements of length, and equations of coordinate surfaces are tabulated for various orthogonal coordinate systems.

These can be written A = uˆi Ai and B = uˆ j B j in shortened tensor notation, such as in Eq. 2-3), where the summation from one to three is suppressed. 4-13) Applying Eq. 4-12) results in six of the above nine terms going to zero, namely the off-diagonal terms, and the diagonal unit vector dot products equaling unity. Therefore, only the diagonal terms of Eq. 4-14) i i =1 which is, of course, a scalar. 4-15) where the six off-diagonal zeros and the three diagonal ones from the Kronecker delta of Eq.

4 Partial derivative of a vector function By the time students in the physical sciences or engineering enter upper-division courses (junior and senior years of a bachelor’s program), they will have been exposed to the partial derivative. However, this introduction was invariably done in Cartesian coordinates with Cartesian-coordinate examples. This was fine when the partial derivative being explained was taken on a scalar function. However, problems can arise if the partial derivative is taken of a vector function (or of any tensor of rank greater than zero) and the physical problem leads naturally into curvilinear coordinates, such as cylindrical coordinates.