Lecture 08: Units and Sample Calculations

Overview

Steven M. Wise reviews tensor-calculus tools for continuum mechanics, beginning with the localization theorem and emphasizing that quantities combined in equations must match in both mathematical type and physical units. Worked examples apply gradients, divergence, curl, and the Laplacian to temperature and velocity fields, then verify the divergence theorem and Stokes' theorem using rectangular, spherical, flat-disc, and paraboloid domains.

Key takeaways

Continuity is essential to the localization theorem: if a continuous field has zero integral over every open patch, it must vanish pointwise.
Physical-unit checks constrain continuum-mechanics models: heat flux in J/(m²·s) produces a divergence and source in J/(m³·s), while velocity divergence has units s⁻¹.
Divergence measures local outward flux per unit volume: sources have positive divergence, sinks negative divergence, and constant flows have zero divergence.
Curl measures circulation per unit area; for the planar rigid rotation w = ω(−x₂e₁ + x₁e₂), curl w = 2ωe₃ and angular velocity is half the curl.
The heat-equation diffusivity must have units m²/s because the Laplacian of temperature has units K/m² while the time derivative has units K/s.
The divergence and Stokes theorem examples show that boundary and interior calculations agree, and Stokes' result depends on the shared boundary and orientation rather than the surface's shape.

Chapters

0:00 Localization Theorem: Zero Integrals on Every Patch Imply a Zero Field

For a continuous scalar field φ on an open set B, if its integral over every open subset ω of B is zero, then φ is zero throughout B.
The proof assumes φ(y) is positive at some point, then uses continuity to find a neighborhood where φ exceeds a positive constant, making that neighborhood's integral positive.
The same reasoning applies component by component to continuous vector and tensor fields.

6:17 Flux, Source, and Matching Units in Balance Laws

A local balance law combines a time derivative, the divergence of a flux Q, and a source S.
If the integral of S minus div Q is zero on every patch, the localization theorem gives S = div Q pointwise, assuming continuity.
For heat transport, flux has units J/(m²·s), so its divergence and the matching source have units J/(m³·s).
Terms can be added only when they have both the same mathematical type and the same physical units.

9:44 Orthogonal Coordinates, Unit Vectors, and Derivative Dimensions

Wise works in a three-dimensional orthogonal frame with coordinates x₁, x₂, x₃ and unit vectors e₁, e₂, e₃.
Coordinates represent fixed points in space, while vectors have magnitude and direction and can be translated without changing their value.
With coordinates measured in length L, a spatial derivative divides a field's units by L; a second derivative divides by L².
Unit vectors are treated as dimensionless, and integrating a field over volume multiplies its units by L³.

18:55 Temperature Gradient: A Worked Units and Direction Example

For a temperature field with constant, linear, and quadratic coordinate terms, the coefficients must carry units K, K/m, and K/m² respectively.
Differentiating the field gives a gradient with units K/m and nonzero components only in directions on which temperature depends.
At any point, the gradient of a scalar temperature field points toward its greatest local increase.

23:32 Velocity Gradient and Divergence for a Diagonal Flow

The example velocity field has components proportional to x₁, −x₂, and x₃, with coefficients a and b measured in s⁻¹.
Its gradient is diagonal because each velocity component depends only on its corresponding coordinate.
Taking the trace gives div v = b; both the velocity gradient and divergence have units s⁻¹.

28:54 Divergence as Outward Flux per Unit Volume

The divergence theorem relates the boundary flux through a small ball to the volume integral of div v inside it.
As the ball around a point shrinks, divergence represents the local outward volume flux per unit volume.
Positive divergence indicates a source, negative divergence a sink, and zero divergence a divergence-free flow; a constant velocity field is an example.

32:40 Expansion and Swirling Examples: Divergence of Two Velocity Fields

For v = γ(x₁e₁ + x₂e₂ + x₃e₃), the divergence is 3γ, positive everywhere when γ is positive.
The planar swirling field w = ω(−x₂e₁ + x₁e₂) has zero divergence because each component's derivative along its own coordinate is zero.
Both γ and ω have units s⁻¹, consistent with velocity components formed by multiplying them by position.

36:46 Curl as Circulation per Unit Area

Stokes' theorem connects the circulation around a shrinking closed curve to the surface integral of curl through the enclosed patch.
In the small-patch limit, curl measures local circulation per unit area and encodes the axis and direction of rotation.
The swirling field has curl 2ωe₃, so its rotation points out of the plane; its rigid-rotation angular velocity is half the curl.

42:40 Temperature Laplacian and the Diffusion Coefficient

For T = T* + c(x₁² + x₂² + x₃²), the Laplacian is 6c because each of the three second derivatives contributes 2c.
If temperature is measured in kelvin, c has units K/m² and the Laplacian has units K/m².
In the heat equation, diffusivity must have units m²/s so that diffusivity times the Laplacian matches the temperature time derivative in K/s.

45:34 Scalar-Vector Product Rules and a Divergence Calculation

Multiplying a scalar field φ by a vector field v produces another vector field; a dimensionless φ leaves the velocity units unchanged.
For the example fields, Wise calculates div(φv) directly and checks it against the product rule div(φv) = ∇φ·v + φ div v.
Each term on the product-rule right-hand side is a scalar with units s⁻¹, matching the divergence of the velocity field.

51:08 Curl Product Rule: Checking Components and Units

The curl product rule is curl(φv) = ∇φ × v + φ curl v.
The example forms φv first, then differentiates its components and compares the result with the two product-rule terms.
The resulting components have units of velocity per length, equivalent to s⁻¹, and the two methods produce the same components.

53:54 Two-Dimensional Divergence Theorem on a Rectangle

Wise tests the theorem on a 2L-by-L rectangle with v = αx₁²e₁ + x₁x₂e₂.
The divergence is 3αx₁, and integrating it over the rectangle gives 3αL³.
Evaluating the four boundary fluxes gives the same total; α has units 1/(m·s), making both results m²/s.

58:50 Three-Dimensional Divergence Theorem on a Ball

For the radial field v = βx on a ball of radius R, div v = 3β, where β has units s⁻¹.
The volume integral is 3β times the ball's volume, yielding 4πβR³.
On the sphere, x = Rn̂, so v·n̂ = βR; multiplying by the sphere area 4πR² gives the same flux, with units m³/s.

1:01:41 Stokes' Theorem on a Flat Disc and a Paraboloid

A vortex field with curl κe₃ is integrated around a circular boundary of radius R, oriented counterclockwise for an upward normal.
The boundary circulation is πκR²; integrating curl over the flat disc gives the same result because curl is constant and aligned with the normal.
Replacing the flat disc with a paraboloid sharing the same boundary preserves the Stokes integral; the parameterized surface calculation is independent of its height.

Transcript excerpt

0:04 Welcome back to continue mechanics. Um we're still in chapter 2 talking about tensor calculus. And as promised today, we'll take some time to really reflect on all the definitions that we've given in this chapter. Before we do that though, we'll state one more result called the localization theorem. It's one that we use quite a lot. It's not It it is fundamental but it's not um completely it's not very deep I would say here it is. So consider an open set B in uklidian space and a continuous function that's defined on that open set and let omega denote an arbitrary open

0:56 set of B. The idea is this should be wholly contained inside of B. And so well I guess that's all I need to say about that. If we find out that for every such set omega the integral of phi is equal to zero then that function has to be zero everywhere in B. So the idea is this. You have a set B and I should think about taking a bunch of smaller sets omega open set and integrating over every patch of that size.

1:44 If I find that after integrating over every patch of that size I get zero then it has to be that the function itself is zero. Now there is one thing that um I didn't emphasize but probably should and that is we need uh phi to be a continuous function for this to be true. So here's the proof. We're going to assume that 2.1 is true but 2.2 is false. So let's assume that this is true but this is not necessarily true to arrive at a contradiction. So we're going to assume that every such integral is equal to zero but there's a point where phi is not equal to zero.

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Summary, takeaways, and chapters were generated by AI from the video's transcript and may contain errors. The video belongs to its creator, Steven M. Wise.