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Lecture Overview

• Instructor: Chase Reuter
• University: University of Arizona
• Date: September 2, 2026
• Purpose: Lectures may be audio recorded for the purpose of a student’s individual study as part of an approved academic accommodation.

Lecture Goals

• Multi-index Notation: Discussion will include topics not found in the text.
• Section 4.4 Topics:
  • Definitions:
  • Gradient: $\nabla f$
  • Divergence: $\nabla \cdot F$
  • Curl: $\nabla \times F$
  • Physics Meaning
  • Connection to k-forms
  • Connection to Traditional Methods (in $\mathbb{R}^3$)
• Section 4.6 Topics:
  • Exact/Closed k-forms
  • Implications
• Section 4.7 Topics (Time permitting):
  • Pullback of a k-form

Recap of Last Class

• Key Concepts:
  • Divergence: $\nabla \cdot F$
  • Curl: $\nabla \times F$
  • Gradient: $\nabla f$
• Terms Explained:
  • (Divergence) Concept relates to "flow into/out of a neighborhood". Symbolized as the sum of infinitesimal volume change in a directional field.
  • (Curl) Represents the tendency of a vector field to induce rotation.
  • (Gradient) Indicates the direction of maximum increase of a function.
• Identities:
  • $\nabla \cdot F \sim d\omega_2$
  • $\nabla \times F \sim d\omega_1$
  • $\nabla f \sim df$
• Notable Equations:
  • Incompressible condition: $\nabla \cdot F = 0$
  • Curl-Free condition: $\nabla \times F = 0$
• Associations:
  • $\omega<em>1 = f</em>1 dx<em>1 + f</em>2 dx<em>2 + f</em>3 dx_3$
  • $\omega<em>2 = f</em>1 dx<em>2 \wedge dx</em>3 - f<em>2 dx</em>1 \wedge dx<em>3 + f</em>3 dx<em>1 \wedge dx</em>2$

Definitions of Exactness

• Exact k-forms: A k-form $\omega$ on $U \subset \mathbb{R}^n$ is exact if there exists a (k - 1)-form $\nu$ such that $\omega = d\nu$.
• Closed k-forms: A k-form $\omega$ on $U \subset \mathbb{R}^n$ is closed if $d\omega = 0$.

Fundamental Results on Exactness

• Theorems:
  • (V1): For a k-form $\omega$ defined on all of $\mathbb{R}^n$, it is exact if and only if it is closed.
  • (V2): For a k-form $\omega$ defined on an open ball $B \subset \mathbb{R}^n$, it is exact if and only if it is closed.

Checking Exactness

• Two-forms on $\mathbb{R}^4$:
  • Example 1: $\omega_1 = x^2 dx \wedge dy + 2yx dx \wedge dz + 3 \cos(2z) dz \wedge dw$
  • Result: $d\omega_1 = 0 + 2y dy \wedge dx \wedge dz + 0$
  • Example 2: $\omega_2 = (\cos(2xy) + e^{-w}) dx \wedge dy + (xe^{-w} + w^2) dy \wedge dw + z^{10} dy \wedge dz$
  • Result: $d\omega_2 = ( -e^{-w} dw) \wedge dx \wedge dy + (e^{-w} dx) \wedge dy \wedge dw = 0$
• Conclusion:
  • By Poincare's theorem (V1), both forms are determined to be exact, implying a corresponding (k-1)-form exists where $\omega = d\nu$ and that it solves a system of PDEs.

Implications and Comparisons in Geometry

• One-forms and Collinearity:
  • A one-form $\omega$ on $\mathbb{R}^3$ is exact if and only if its associated vector field is conservative, where $F = \nabla f$ for some potential function $f$.
• Two-forms:
  • A two-form $\omega$ on $\mathbb{R}^3$ is exact if it has a vector field potential, meaning if $\omega = d\nu$ and $G \sim \nu$; hence $\omega = d\nu \sim \text{curl}(G) = \nabla \times G = F$.

Closed Forms and Conservative Fields

• One-forms: A one-form $\omega$ is closed if and only if its associated vector field $F$ is curl-free.
  • Notation: $0 = d\omega \sim \text{curl}(F) = \nabla \times F = 0$.
• Two-forms: A two-form $\omega$ is closed if and only if its associated vector field is divergence-free.
  • Notation: $0 = d\omega \sim \nabla \cdot F = 0$.

Further Implications on Non-exactness

• One-forms: A one-form $\omega$ that is not closed is not exact.
  • Equivalently, a vector field $F$ that is not curl-free (i.e., $\nabla \times F = 0$) is not conservative, indicating the absence of a potential function.
• Contrapositive: This principle leads to the corollary: "If a one-form is exact, then it is closed".
• Two-forms: Similarly, a two-form $\omega$ that is not closed is not exact, and a vector field $F$ associated with a two-form that is not divergence-free (i.e., $\nabla \cdot F = 0$) does not admit a vector potential.

Poincare's Lemma Applications

• One-forms: Poincare’s lemma indicates that if $F$ is defined with continuous first-order partial derivatives on all of $\mathbb{R}^3$ or any open simply connected subset, then it is conservative if and only if curl-free.
• Two-forms: The lemma states for a two-form that it has a vector potential if and only if it is divergence-free under similar conditions.

Pullbacks on 1-forms

• Definition: For pullbacks $\varphi : V \subset \mathbb{R}^n \to U \subset \mathbb{R}^n$ on a one-form $\omega = \sum<em>{J} f</em>J dx^J$, they must satisfy specific operations:
  • $\varphi^<em>(\omega + \nu) = \varphi^</em>\omega + \varphi^*\nu$
  • $\varphi^<em>(f \omega) = (\varphi^</em>f)(\varphi^*\omega)$
  • $\varphi^<em>(df) = d(\varphi^</em>f)$

General Pullback Rule for 1-forms

• Applying these principles leads to the formula for pullbacks of 1-forms:
  • $\varphi^<em>\omega = \sum<em>{i=1}^{n} (\varphi^</em>f</em>i)(\varphi^*dx_i)$
• Wedge Product Considerations:
  • The criteria can be adjusted for k-forms, revealing that the wedge product simplifies to multiplication when one factor is a 0-form.

Pullbacks on General k-forms

• The basic operation relations for pullbacks are extended to k-forms:
  • $\varphi^<em>(\omega + \nu) = \varphi^</em>\omega + \varphi^*\nu$
  • $\varphi^<em>(\nu \wedge \omega) = (\varphi^</em>\nu) \wedge (\varphi^*\omega)$
  • $\varphi^<em>(df) = d(\varphi^</em>f)$
• Explicit Pullback Formula:
  • For a basic k-form, we arrive at $\varphi^<em>(dx^I) = (\varphi^</em>dx<em>{i</em>1}) \wedge \cdots \wedge (\varphi^*dx<em>{i</em>k})$.

Example Calculation for Pullbacks

• Specific Case: Consider the pullback given by the transformation $T(x, y, z) = \begin{pmatrix} x + y \sqrt{2} \ \ x - y \sqrt{2} \ \ z \end{pmatrix}$.
  • Calculate the pullback of $\omega = dy \wedge dz - dx \wedge dz + dx \wedge dy$:
  • The pullback results in:
  • $T^*\omega = -\sqrt{2} dx<em>2 \wedge dx</em>3 - \sqrt{2} dx<em>1 \wedge dx</em>2$.

The Jacobian in Pullbacks

• Jacobian Definition: For a smooth function $\varphi : V \to U$ in open $U, V \subset \mathbb{R}^n$.
  • Denote the Jacobian as $J_{\varphi}$ or $D\varphi$, forming an n x n matrix of first partial derivatives:
  • $D\varphi = \begin{pmatrix} \frac{\partial \varphi<em>1}{\partial t</em>1} & \cdots & \frac{\partial \varphi<em>1}{\partial t</em>n} \ \vdots & \ddots & \vdots \ \frac{\partial \varphi<em>n}{\partial t</em>1} & \cdots & \frac{\partial \varphi<em>n}{\partial t</em>n} \end{pmatrix}$
  • Jacobian Determinant: Often denoted $\text{det} J_{\varphi}$, it gathers importance in transformations.

Top Form Pullbacks

• Definition: For a top form $\omega = f dx^1 \wedge \cdots \wedge dx^n$ and smooth function $\varphi: V \to U$:
  • Results in: $\varphi^<em>\omega = (\varphi^</em>f)(t)(\text{det} J_{\varphi}) dt^1 \wedge \, \cdots \wedge dt^n$.
• Simplification for 2D Case:
  • We use $d\varphi<em>1 \wedge d\varphi</em>2 = (\varphi<em>1^1 dt^1 + \varphi</em>1^2 dt^2) \wedge (\varphi<em>2^1 dt^1 + \varphi</em>2^2 dt^2)$
  • Collect products and derive combinations, yielding equivalence with the determinant form.

Observations on Determinants and Higher Dimensions

• To check forms in diverse cases, we compute determinants through expansion along rows or columns, bolstering proofs.
• It remains essential to relate these to the reordering argument within higher-dimensional contexts, wrapping the theoretical framework around practical narrative examples.