Express the cylindrical unit vectors 𝐬̂, ϕ̂, 𝐳̂ in terms of 𝐱̂, 𝐲̂, 𝐳̂ (that is, derive Eq. 1.75

Express the cylindrical unit vectors 𝐬̂, ϕ̂, 𝐳̂ in terms of...

Express the cylindrical unit vectors $\hat{\mathbf{s}}, \hat{\boldsymbol{\phi}}, \hat{\mathbf{z}}$ in terms of $\hat{\mathbf{x}}, \hat{\mathbf{y}}, \hat{\mathbf{z}}$ (that is, derive Eq. 1.75 ). "Invert" your formulas to get $\hat{\mathbf{x}}, \hat{\mathbf{y}}, \hat{\mathbf{z}}$ in terms of $\hat{\mathbf{s}}, \hat{\boldsymbol{\phi}}, \hat{\mathbf{z}}$ (and $\phi$ ).

Key Concepts

Basis Transformation

Basis transformation involves expressing the unit vectors of one coordinate system in terms of those of another. This is achieved through the use of trigonometric relationships derived from the geometric connection between the coordinate systems, allowing one to ‘invert’ the transformation to convert back and forth between cylindrical and Cartesian representations.

Unit Vectors

Unit vectors are vectors of length one that indicate direction along the axes of a coordinate system. They form the basis for expressing other vectors, and transforming between different sets of unit vectors is essential for converting vector components between coordinate systems.

Cartesian Coordinate System

This system uses three orthogonal coordinates (x, y, z) with fixed unit vectors, making it a standard framework for describing spatial positions and vector components. Its simplicity in representation and calculation makes it a common reference frame in vector analysis.

Cylindrical Coordinate System

This system represents points in space using a radial distance, an angular coordinate, and a height (typically denoted as (s, ?, z)). It is particularly useful for problems with rotational symmetry around an axis, and its corresponding unit vectors vary with position, as they depend on the angular coordinate ?.

Transcript

00:01 Express the cylindrical unit vectors s -cap, phi -cap, z -cap, in terms of x -cap, y -cap, z -cap, and invert the formulas to get x -cap, y -cap, z -cap in terms of s -cap, phi -cap and z -cap.

00:18 A point p in the cylindrical coordinate system is represented by p, s, comma, 5, z, your s is the distance of point p from the z -axis, phi, phi, phi, phi, phi, phi, phi, phi is the azimethal angle and z is the coordinate of point p on the x xx.

00:35 Considering the following figure, the transformation from cartesian coordinates to the cylindrical coordinates is given by x equals s cos phi, y equals s sine 5, and z equals z.

00:50 The unit vectors in the cylindrical coordinates s cap equals cos -fi x -cap plus sine -fi y -cap 5 -cap 5 -cap equals minus of sine -fi x -cap plus cos -fi -y -cap and z -cap equals x -cap an infinitesimal displacement along as while keeping phi and z as constant so d l x equals dxxx cap plus d .y y cap plus dz z cap.

01:26 Your dx, dx, d, d, z represent the displacement along x, y, and z axis in the cartesian system.

01:32 D differentiate the transformation where dx equal cos phi d .s, dy equals sine phi ds, and dz equals 0.

01:44 Substituting these values into the equation dlx equal dlx equal cos 5 dsx cap plus sine 5 ds y cap plus 0 z cap and the equation will become ds taking common cost 5 x cap plus sine 5 ds cap plus sine 5 y cap but we know that d l x is equals to d s into s cap comparing the equations one and two we get the first expression of s cap equals cos 5 x cap plus sine 5 y cap now keeping s and z as constant values finding dl -5 equals s, d -5, phi -cap, where dl -5 can also be written as d -x, x, kappa plus d -y, y -cap plus d -z -z -cap.

03:09 Dens.

03:10 Diffensiate the transformation equation in the form of 5, d -x is minus s -s -s -s -s -s -5 -d -5, d -y equals s -c -cos -5, d -5, and d -z -e -5.

03:26 Equals 0...

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