Derive and validate the gradient formula for a physical vector field in orthogonal curvilinear coordinates using scale factors, basis vectors, and Christoffel symbols | Step-by-Step Solution

Understanding the Gradient in Curvilinear Coordinates

What We're Solving:

We need to derive the formula for the gradient of a vector-valued function in orthogonal curvilinear coordinates (like cylindrical or spherical coordinates) using tensor calculus concepts. This involves understanding how scale factors, basis vectors, and Christoffel symbols work together to properly describe how vectors change in curved coordinate systems.

Our strategy involves:

• Setting up the coordinate system with proper basis vectors
• Understanding how scale factors account for coordinate stretching
• Using Christoffel symbols to handle the changing geometry
• Applying tensor calculus rules to derive the gradient formula

Step-by-Step Solution:

Step 1: Establish the Coordinate System Start with orthogonal curvilinear coordinates (u¹, u², u³). The key insight is that at each point, we have:

• Scale factors h₁, h₂, h₃ that tell us how "stretched" each coordinate direction is
• Unit basis vectors ê₁, ê₂, ê₃ that point along coordinate curves
• These basis vectors can change direction from point to point (unlike Cartesian!)

Step 2: Express the Vector Field Write your vector field F in terms of the local basis: F = F¹ê₁ + F²ê₂ + F³ê₃

The components F¹, F², F³ are the "amounts" of the vector in each coordinate direction.

Step 3: Understand Why We Need Christoffel Symbols When we take derivatives in curvilinear coordinates, the basis vectors themselves change! The Christoffel symbols Γᵢⱼᵏ measure exactly how much each basis vector changes as we move in different directions.

For orthogonal systems, these symbols have a special form involving the scale factors and their derivatives.

Step 4: Apply the Gradient Definition The gradient of a vector field is a tensor that tells us how each component of the vector changes in each coordinate direction. In tensor notation: (∇F)ᵢⱼ = ∂Fⁱ/∂uʲ + Γᵢₖⱼ Fᵏ

The first term is the "obvious" derivative of components, and the second term corrects for the changing basis vectors.

Step 5: Substitute Orthogonal Coordinate Expressions For orthogonal coordinates, the Christoffel symbols simplify significantly. Most are zero, and the non-zero ones have specific patterns involving derivatives of the scale factors.

Step 6: Work Out the Components Systematically compute each component of the gradient tensor, being careful about:

• Which Christoffel symbols are non-zero
• How scale factors appear in derivatives
• The geometric meaning of each term

The Answer:

The gradient of vector field F = F¹ê₁ + F²ê₂ + F³ê₃ in orthogonal curvilinear coordinates is:

∇F = Σᵢⱼ [(1/hⱼ)(∂Fⁱ/∂uʲ) + (correction terms involving Christoffel symbols)] êᵢ ⊗ êʲ

The specific form depends on your coordinate system, but the pattern involves:

• Derivatives of components divided by scale factors
• Correction terms that account for basis vector changes
• Each term having clear geometric interpretation

Validation: Check your result by:

• Verifying it reduces to the Cartesian formula when hᵢ = 1
• Confirming the result matches known formulas for cylindrical/spherical coordinates
• Checking that the tensor transforms properly under coordinate changes

Memory Tip:

Think "derivative + geometry correction"! The gradient in curvilinear coordinates is always the "obvious" derivative of components, plus correction terms that account for the curved geometry. The Christoffel symbols are your "geometry correction factors" - they encode how the coordinate system itself is bending and stretching.

Remember: tensor calculus isn't about making things complicated - it's about being precise when geometry gets interesting! 🌟