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Vector field to cylindrical at point

Short answer

A vector field assigns a vector to each point in space. At P(2,30°,6), the cylindrical components are A=0.316 a_ρ + 0.949 a_z.

Define a vector field. A field vector is given by A = [1/√(x² + y² + z²)] (x a_x + y a_y + z a_z). Transform this vector into cylindrical coordinate system at point (2, 30°, 6).

Vector field: A vector field is a function that assigns a vector (magnitude and direction) to every point in a region of space. Examples: electric field E(x, y, z), velocity of flowing water, gravitational field. (A scalar field assigns only a number, such as temperature.)

Transformation

Given: A = (x a_x + y a_y + z a_z) / √(x² + y² + z²) at the point (2, 30°, 6), taken as cylindrical coordinates (ρ = 2, φ = 30°, z = 6).

Cartesian position of the point:

x = ρ cosφ = 2 cos30° = 1.732,   y = ρ sinφ = 2 sin30° = 1,   z = 6
r = √(x² + y² + z²) = √(3 + 1 + 36) = √40 = 6.325

Cartesian components of A:

A_x = x/r = 0.2739,   A_y = y/r = 0.1581,   A_z = z/r = 0.9487

Cartesian to cylindrical:

A_ρ = A_x cosφ + A_y sinφ = 0.2739(0.8660) + 0.1581(0.5) = 0.2372 + 0.0791 = 0.3162
A_φ = −A_x sinφ + A_y cosφ = −0.2739(0.5) + 0.1581(0.8660) = −0.1369 + 0.1369 = 0
A_z = 0.9487

Check: A is a unit vector (it points radially outward from the origin): 0.3162² + 0.9487² = 0.1 + 0.9 = 1 ✓. In general A_ρ = ρ/√(ρ² + z²) = 2/√40 and A_z = z/√(ρ² + z²) = 6/√40.

Answer: A = 0.316 a_ρ + 0.949 a_z

✓ Answer checked by Claude Opus 5.5 (high effort)

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