Analysis of the Curve and Tangent in Cartesian Coordinates
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Cartesian coordinates, curve, tangent, vector normal, plane, area of parallelogram, volume of parallelepiped, axial symmetry
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Abstract
This document provides a detailed analysis of the curve and tangent in Cartesian coordinates, including the expression of the field in Cartesian coordinates, the determination of the vector normal to the plane, and the calculation of the area of the parallelogram and the volume of the parallelepiped. The results obtained can be extended to the entire curve by performing an axial symmetry with respect to the vertical axis passing through the origin.
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[...] The line may be represented by . ; The line may be represented by . ; The line may be represented by . On a bien so there is a function such as . Exercise 7 and for variant of à . In Cartesian coordinates: so and . We have then: It is about circle of centre and radius . Or (we can demonstrate this result using Euler's formula) So: . so this field can derive from a potential such as . [...]
[...] From where the following system:. We see that an obvious solution is from where . The equation of the plane is therefore: . As the point belongs to the plan so we have: so . The equation of the plane is therefore . So we are looking for the point , projection de on the level as if . We have then: so and . The point a then for coordinates . so we have the following system: . 10) and . [...]
[...] - so for the curve passes through the point and there for tangent . - so for the curve passes through the point and there for tangent . The coordinates of in the database so are . so . We deduce that: . This corresponds to considering the right-hand side of the curve. The results obtained for this part can be extended to the entire curve by performing an axial symmetry with respect to the vertical axis passing through the origin. [...]
