Curvilinear Canvas Slides Curvilinear Canvas
Lesson 1: Basis Vectors and Local Frames
Vector Fields and Tensor Analysis
The Problem with Grids
Cartesian coordinates \((x, y, z)\) are global and uniform. Their basis vectors \(\mathbf{\hat{i}}, \mathbf{\hat{j}}, \mathbf{\hat{k}}\) are constant.
The Reality Check:
Most physical systems—atoms, stars, fluid flows—exhibit symmetries that Cartesian grids fail to capture efficiently.
Coordinate Dependence
Boundary conditions are easier to satisfy.
Symmetry reduces the degrees of freedom.
Trade-off: Basis vectors now vary with position.
General Curvilinear Coordinates
Let \(u^1, u^2, u^3\) be a set of general coordinates. The transformation from Cartesian space is given by:
\(\mathbf{r} = \mathbf{r}(u^1, u^2, u^3)\)
Holonomic Basis
Natural basis vectors tangent to the coordinate lines:
\(\mathbf{e}_i = \frac{\partial \mathbf{r}}{\partial u^i}\)
Scale Factors
The magnitude of the basis vectors:
\(h_i = |\mathbf{e}_i|\)
Example: Cylindrical Coordinates
Transformations:
\(x = \rho \cos \phi\)
\(y = \rho \sin \phi\)
\(z = z\)
Calculate \(\mathbf{e}_\phi\):
The basis vector tangent to the \(\phi\)-direction.
\(\mathbf{e}_\phi = \frac{\partial \mathbf{r}}{\partial \phi} = -\rho \sin \phi \mathbf{\hat{i}} + \rho \cos \phi \mathbf{\hat{j}}\)
Magnitude \(h_\phi = \rho\). Note that the basis depends on the radius!
Visualization: Basis vectors \(\mathbf{\hat{e}}_\rho, \mathbf{\hat{e}}_\phi, \mathbf{\hat{e}}_z\) rotating with position.
Orthogonality
A coordinate system is orthogonal if the basis vectors at any point are mutually perpendicular.
\(\mathbf{e}_i \cdot \mathbf{e}_j = 0\) for \(i \neq j\)
In general curvilinear systems (like in General Relativity), this is not always the case.
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Orthogonal: Cylindrical, Spherical, Ellipsoidal.
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Non-Orthogonal: Skewed grids, Oblique coordinates.
Mastering the Basis
The choice of basis is the choice of perspective. In Lesson 2, we will see how these components transform under change of basis.
Practice calculations
Review transformations
Read Chapter 1.2
Coordinate Shift Workshop Coordinate Shift Workshop
Lesson 1: Basis Vectors and Local Frames
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Technical Briefing
In general curvilinear coordinates \( (u^1, u^2, u^3) \), the holonomic (natural) basis vectors are defined by the partial derivatives of the position vector: \(\mathbf{e}_i = \frac{\partial \mathbf{r}}{\partial u^i}\). The scale factors are the magnitudes of these vectors: \(h_i = |\mathbf{e}_i|\). The unit basis vectors (physical basis) are \(\mathbf{\hat{e}}_i = \frac{\mathbf{e}_i}{h_i}\).
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Parabolic Coordinate Transformation
Consider the 2D parabolic coordinate system \( (\xi, \eta) \) defined by:
\(x = \xi\eta\)
\(y = \frac{1}{2}(\eta^2 - \xi^2)\)
A. Calculate the holonomic basis vectors \(\mathbf{e}_\xi\) and \(\mathbf{e}_\eta\) in terms of the Cartesian basis vectors \(\mathbf{\hat{i}}\) and \(\mathbf{\hat{j}}\).
B. Determine the scale factors \(h_\xi\) and \(h_\eta\) for this system.
C. Prove that this coordinate system is orthogonal.
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Vector Field Decomposition
A vector field is given by \(\mathbf{V} = y\mathbf{\hat{i}} - x\mathbf{\hat{j}}\). We wish to express this vector in the parabolic basis.
A. Write \(\mathbf{V}\) in terms of \(\xi\) and \(\eta\).
B. Express \(\mathbf{V}\) as a linear combination of the unit basis vectors \(\mathbf{\hat{e}}_\xi\) and \(\mathbf{\hat{e}}_\eta\).
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Reflective Analysis
Think about the dependence of your results on position.
Why are these "local" basis vectors? Describe what happens to the orientation and magnitude of \(\mathbf{e}_\xi\) as we move along a line of constant \(\eta\).
Advanced Vector Analysis • Graduate Physics • Module 1.1
Component Dualism Slides Component Dualism
Lesson 2: Covariant vs. Contravariant Transformations
Vector Fields and Tensor Analysis
Stretch and Shrink
Imagine you stretch your coordinate grid by a factor of 2.
What happens to the vector?
The basis vectors grow, but the components must shrink to keep the physical vector unchanged.
CONTRAVARIANT (Upper Indices)
Transform "against" the change in basis.
COVARIANT (Lower Indices)
Transform "with" the change in basis.
The Einstein Convention
Indices appearing once in an upper position and once in a lower position are implicitly summed over.
Standard Notation
\(\mathbf{V} = \sum_{i=1}^n V^i \mathbf{e}_i\)
Einstein Notation
\(\mathbf{V} = V^i \mathbf{e}_i\)
"I have made a great discovery in mathematics; I have suppressed the summation sign every time that the index occurs twice..." — Albert Einstein
Contravariant Rule
Vector components \(V^i\) are contravariant if they transform under a coordinate change \(x^i \to \bar{x}^j\) as:
\(\bar{V}^j = \frac{\partial \bar{x}^j}{\partial x^i} V^i\)
Example: Velocity, Displacement, Current Density.
Transformation Matrix
The partial derivative \(\frac{\partial \bar{x}^j}{\partial x^i}\) represents the Jacobian of the transformation.
Notice: The indices "cancel" like fractions to leave the correct index on the left.
Covariant Rule (Covectors)
The Dual View
Components that transform using the inverse Jacobian. They represent linear functionals on the vector space.
Key Example: The Gradient \(\nabla \phi\).
Components \(V_i\) are covariant if they transform as:
\(\bar{V}_j = \frac{\partial x^i}{\partial \bar{x}^j} V_i\)
Notice the placement of indices in the partial derivative!
The Taxonomy of Tensors
Property Contravariant Covariant Index Position Upper (\(V^i\)) Lower (\(V_i\)) Transformation \(\frac{\partial \bar{x}^j}{\partial x^i}\) \(\frac{\partial x^i}{\partial \bar{x}^j}\)
Index Mastery Challenge Index Mastery Challenge
Lesson 2: Einstein Notation and Transformation Rules
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1
Summation Mechanics
Expand the following expressions using the Einstein summation convention (assume \(n=3\)):
A) \( A_i B^i \) Scalar Product
B) \( T^i_{\phantom{i}j} V^j \) Tensor Contraction
C) \( \frac{\partial V^i}{\partial x^i} \) Divergence (approx.)
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The Jacobian Bridge
Consider a coordinate transformation from \( x^i \) to \( \bar{x}^j \).
A. If \( T^i \) is a contravariant vector, write the explicit formula for its component \( \bar{T}^j \) in the new coordinate system.
B. Prove that the gradient of a scalar field \( \phi \), denoted by \( \partial_i \phi = \frac{\partial \phi}{\partial x^i} \), transforms as a covariant vector.
Hint: Use the multivariable chain rule.
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Structural Integrity
A vector is a physical object that exists independently of the coordinate system. Explain, in your own words, why we need both covariant and contravariant components to describe this object in non-orthogonal or curved spaces.
True or False: If two vectors are equal in one coordinate system, they must be equal in all coordinate systems. Justify your answer using the transformation laws.
Advanced Vector Analysis • Graduate Physics • Module 2.1
Metric Mapping Slides Metric Mapping
Lesson 3: The Geometry of Space
Vector Fields and Tensor Analysis
Measuring the Warp
In a flat Cartesian world, distance is simple: \(d^2 = dx^2 + dy^2 + dz^2\).
But what if the world is warped?
How do we define distance on a sphere? Or near a black hole? Or inside a stressed elastic material?
The Metric Tensor is the mathematical tool that encodes the "intrinsic" geometry of the space.
The Metric Tensor \(g_{ij}\)
The metric tensor is defined by the inner products of the basis vectors:
\(g_{ij} = \mathbf{e}_i \cdot \mathbf{e}_j\)
The Line Element:
\(ds^2 = g_{ij} dx^i dx^j\)
Einstein summation convention applies!
Symmetry:
\(g_{ij} = g_{ji}\)
The metric is always a symmetric tensor of rank 2.
Musical Chairs with Indices
The metric tensor \(g_{ij}\) and its inverse \(g^{ij}\) allow us to switch between covariant and contravariant descriptions.
Lowering an index:
\(V_i = g_{ij} V^j\)
Raising an index:
\(V^i = g^{ij} V_j\)
Why do this?
Contravariant components (\(V^i\)) are the "natural" vector components. Covariant components (\(V_i\)) are the "projections" onto the basis vectors.
In Cartesian space, \(g_{ij} = \delta_{ij}\), so the components are identical. In curved space, they are distinct entities!
The Generalized Dot Product
The scalar product of two vectors \(\mathbf{A}\) and \(\mathbf{B}\) can be computed in multiple equivalent ways:
Mixed Components
\(A^i B_i\)
Metric Form
\(g_{ij} A^i B^j\)
Dual Form
\(A_i B^i\)
This result is a scalar (invariant). It does not change value under coordinate transformations.
Metric and Orthogonality
Orthogonal Basis
The metric tensor is diagonal . All off-diagonal components \(g_{ij} = 0\) for \(i \neq j\).
Non-Orthogonal Basis
The metric tensor has non-zero off-diagonal components. This indicates "skew" in the coordinate system.
Spherical Example
In spherical coordinates \((r, \theta, \phi)\):
\(g_{ij} = \begin{pmatrix} 1 & 0 & 0 \\ 0 & r^2 & 0 \\ 0 & 0 & r^2 \sin^2 \theta \end{pmatrix}\)
Notice the diagonal structure. Spherical coordinates are orthogonal!
Geometric Weaver Workshop Geometric Weaver Workshop
Lesson 3: The Metric Tensor and Index Manipulation
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1
The Metric from Transformations
Consider the 2D transformation from Cartesian coordinates \((x, y)\) to new coordinates \((u, v)\): \[x = u^2 - v^2, \quad y = 2uv\]
A. Find the basis vectors \(\mathbf{e}_u\) and \(\mathbf{e}_v\).
B. Calculate the components of the metric tensor \(g_{ij}\) in matrix form.
C. Determine the line element \(ds^2\) for this system.
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Index Gymnastics
Suppose we have a metric tensor given by \(g_{ij} = \begin{pmatrix} 1 & \alpha \\ \alpha & 1 \end{pmatrix}\) where \(0 < \alpha < 1\).
A. Find the components of the inverse metric tensor \(g^{ij}\).
B. Given a contravariant vector \(V^i = (1, 0)\), calculate the covariant components \(V_i\).
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Length and Angle
In the coordinate system with metric \(g_{ij} = \begin{pmatrix} 1 & \alpha \\ \alpha & 1 \end{pmatrix}\), consider two vectors \(A^i = (1, 0)\) and \(B^i = (0, 1)\).
A. Calculate the magnitude (length) of vectors \(A\) and \(B\). Does the parameter \(\alpha\) affect the length of these basis-aligned vectors?
B. Calculate the angle \(\theta\) between vectors \(A\) and \(B\). Express \(\cos \theta\) in terms of \(\alpha\). What value of \(\alpha\) corresponds to an orthogonal system?
Advanced Vector Analysis • Graduate Physics • Module 3.1
Field Calculus Slides Field Calculus
Lesson 4: Covariant Differentiation & Christoffel Symbols
Vector Fields and Tensor Analysis
Differentiating Vectors
When we differentiate a vector field \(\mathbf{V} = V^i \mathbf{e}_i\), we must apply the product rule:
\(\frac{\partial \mathbf{V}}{\partial x^j} = \frac{\partial V^i}{\partial x^j} \mathbf{e}_i + V^i \frac{\partial \mathbf{e}_i}{\partial x^j}\)
In Cartesian space, \(\partial \mathbf{e}_i / \partial x^j = 0\). But in curvilinear space, the basis vectors change!
The Connection
How do the basis vectors change as we move? We need a way to "connect" adjacent tangent spaces.
Christoffel Symbols \(\Gamma^k_{ij}\)
These symbols encode the change of the basis vectors \(\mathbf{e}_i\) with respect to coordinates \(x^j\):
\(\frac{\partial \mathbf{e}_i}{\partial x^j} = \Gamma^k_{ij} \mathbf{e}_k\)
Calculation from Metric:
\(\Gamma^k_{ij} = \frac{1}{2} g^{kl} (\partial_j g_{il} + \partial_i g_{jl} - \partial_l g_{ij})\)
Property:
Christoffel symbols are NOT tensors. They do not transform according to the tensor rule!
The Covariant Derivative \(\nabla_j\)
To ensure the derivative of a tensor is itself a tensor, we define the covariant derivative :
For a Contravariant Vector \(V^i\):
\(\nabla_j V^i = \partial_j V^i + \Gamma^i_{jk} V^k\)
For a Covariant Vector \(V_i\):
\(\nabla_j V_i = \partial_j V_i - \Gamma^k_{ji} V_k\)
"The covariant derivative corrects the coordinate-based derivative for the 'fictitious' change caused by the curving basis vectors."
Divergence and Curl
Divergence
The divergence is the contraction of the covariant derivative:
\(\text{div } \mathbf{V} = \nabla_i V^i\)
Using metric determinant \(g = \det(g_{ij})\):
\(\nabla_i V^i = \frac{1}{\sqrt{g}} \partial_i (\sqrt{g} V^i)\)
Curl
In 3D, the curl is related to the antisymmetric part of the derivative:
\((\text{curl } \mathbf{V})^i = \epsilon^{ijk} \nabla_j V_k\)
Note: For a covariant vector, the curl involves the simple exterior derivative:
\(W_{ij} = \partial_i V_j - \partial_j V_i\)
Parallel Transport
A vector is parallel transported along a curve if its covariant derivative along that curve is zero.
\(\frac{D V^i}{d\lambda} = \frac{dx^j}{d\lambda} \nabla_j V^i = 0\)
Connection Mechanics Challenge Worksheet Connection Mechanics Challenge
Lesson 4: Covariant Differentiation & Curvilinear Operators
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1
The Geometry of a Plane
Consider standard 2D polar coordinates \((r, \theta)\) where the metric is \(g_{rr}=1\), \(g_{\theta\theta}=r^2\), and \(g_{r\theta}=0\).
A. Calculate all non-zero Christoffel symbols \(\Gamma^k_{ij}\) for this system.
Recall: \(\Gamma^k_{ij} = \frac{1}{2} g^{kl} (\partial_j g_{il} + \partial_i g_{jl} - \partial_l g_{ij})\)
B. Write the explicit expression for the covariant derivative \(\nabla_r V^r\) and \(\nabla_\theta V^\theta\).
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Generalized Divergence
Using your results from Task 1, calculate the total divergence \(\nabla_i V^i = \nabla_r V^r + \nabla_\theta V^\theta\). Does this match the standard formula for \(\text{div } \mathbf{V}\) in polar coordinates?
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Geodesic Logic
Challenge Question:
The geodesic equation is given by: \[\frac{d^2 x^k}{d\lambda^2} + \Gamma^k_{ij} \frac{dx^i}{d\lambda} \frac{dx^j}{d\lambda} = 0\] Using the symbols you calculated for polar coordinates, write out the two coupled differential equations for a geodesic (a straight line) in the plane \((r(\lambda), \theta(\lambda))\).
Physical Interpretation: Why do Christoffel symbols vanish in Cartesian coordinates but not in polar coordinates, even though the space itself (the plane) is flat?
Advanced Vector Analysis • Graduate Physics • Module 4.1
Continuous Force Slides Continuous Force
Lesson 5: The Cauchy Stress Tensor
Vector Fields and Tensor Analysis
Internal Struggle
In particle mechanics, force is a simple vector \(\mathbf{F}\). In a solid or fluid, force is distributed.
The Traction Vector \(\mathbf{T}\):
The force per unit area acting on a surface with normal \(\mathbf{n}\).
The traction depends on the orientation of the surface!
The Cauchy Stress Tensor \(\sigma^{ij}\)
Cauchy's Theorem states that the traction vector \(\mathbf{T}\) is a linear transformation of the normal vector \(\mathbf{n}\):
\(T^i = \sigma^{ij} n_j\)
Physical Meaning:
\(\sigma^{ij}\) is the force in the \(i\)-direction per unit area on a surface with normal in the \(j\)-direction.
Symmetry:
Conservation of angular momentum implies \(\sigma^{ij} = \sigma^{ji}\).
Force Balance
For a continuous medium in static equilibrium, the sum of internal and external forces must vanish.
Generalized Equation:
\(\nabla_j \sigma^{ij} + f^i = 0\)
Note the use of the covariant derivative \(\nabla_j\). This ensures the equation holds even in curved coordinate systems (like a rotating frame or a curved shell).
External Force Density \(f^i\):
Examples: Gravity, electromagnetic fields.
Decomposing the Stress
Normal Stress
The diagonal components \(\sigma^{ii}\) (no sum). These represent tension or compression.
Shear Stress
The off-diagonal components \(\sigma^{ij}\) (\(i \neq j\)). These represent "sliding" forces between layers.
In a fluid at rest: \(\sigma^{ij} = -p g^{ij}\)
Stress is isotropic and purely compressive (pressure).
The Universal Language
Whether it's the bending of a beam, the flow of the ocean, or the curvature of spacetime...
Physics
Geometry
Tensors
"The laws of physics must be invariant under coordinate transformations."
Stress Field Analysis Worksheet Stress Field Analysis
Lesson 5: Equilibrium in Continuous Media
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Scenario: The Pressurized Cylinder
Consider a thick-walled hollow cylinder of inner radius \(a\) and outer radius \(b\), subject to an internal pressure \(P_i\). We model this using cylindrical coordinates \((\rho, \phi, z)\). Due to symmetry, the stress tensor is diagonal: \(\sigma^{\rho\rho}(\rho)\), \(\sigma^{\phi\phi}(\rho)\), and \(\sigma^{zz}(\rho)\).
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Equilibrium in Cylindrical Coordinates
In Lesson 4, we learned that \(\nabla_j \sigma^{ij} = 0\) (ignoring body forces). For the radial direction (\(i=\rho\)), this expands to: \[\partial_\rho \sigma^{\rho\rho} + \Gamma^\rho_{\rho\rho} \sigma^{\rho\rho} + \Gamma^\rho_{\phi\phi} \sigma^{\phi\phi} + \Gamma^\phi_{\phi\rho} \sigma^{\rho\phi} + \dots = 0\]
A. Using the Christoffel symbols for cylindrical coordinates (\(\Gamma^\rho_{\phi\phi} = -\rho, \Gamma^\phi_{\phi\rho} = 1/\rho\), others zero), simplify the equilibrium equation for the radial component.
B. If the stress is related to the pressure such that \(\sigma^{\phi\phi} - \sigma^{\rho\rho} = 2A/\rho^2\), solve for the radial stress \(\sigma^{\rho\rho}(\rho)\).
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Traction and Boundaries
The internal pressure acts at \(\rho = a\). Write the boundary condition for the stress component \(\sigma^{\rho\rho}\) at the inner surface. Hint: Consider the normal vector \(\mathbf{n}\) and the traction vector \(\mathbf{T}\).
At the outer surface \(\rho = b\), the cylinder is stress-free. What does this imply for the value of \(\sigma^{\rho\rho}(b)\)?
3
Physical Integration
A tensor is a physical reality, not just a matrix. Explain how the same internal force state (the stress field) can look completely different if we viewed it from a rotating coordinate system versus the fixed cylindrical system used here.
Final Thought: How does the metric tensor bridge the gap between "geometric curvature" (curvilinear coordinates) and "physical force" (stress)?
Advanced Vector Analysis • Graduate Physics • Module 5.1
