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Gibbs Free Energy

Gibbs Free Energy (G) is a thermodynamic potential that helps predict whether a process will occur spontaneously at constant temperature and pressure. It is defined by the equation:

G=H−TSG = H - TSG=H−TS

where HHH is the enthalpy, TTT is the absolute temperature in Kelvin, and SSS is the entropy. A decrease in Gibbs Free Energy (ΔG<0\Delta G < 0ΔG<0) indicates that a process can occur spontaneously, whereas an increase (ΔG>0\Delta G > 0ΔG>0) suggests that the process is non-spontaneous. This concept is crucial in various fields, including chemistry, biology, and engineering, as it provides insights into reaction feasibility and equilibrium conditions. Furthermore, Gibbs Free Energy can be used to determine the maximum reversible work that can be performed by a thermodynamic system at constant temperature and pressure, making it a fundamental concept in understanding energy transformations.

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Green’S Theorem Proof

Green's Theorem establishes a relationship between a double integral over a region in the plane and a line integral around its boundary. Specifically, if CCC is a positively oriented, simple closed curve and DDD is the region bounded by CCC, the theorem states:

∮C(P dx+Q dy)=∬D(∂Q∂x−∂P∂y) dA\oint_C (P \, dx + Q \, dy) = \iint_D \left( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \right) \, dA∮C​(Pdx+Qdy)=∬D​(∂x∂Q​−∂y∂P​)dA

To prove this theorem, we can utilize the concept of a double integral. We divide the region DDD into small rectangles, and apply the Fundamental Theorem of Calculus to each rectangle. By considering the contributions of the line integral along the boundary of each rectangle, we sum these contributions and observe that the interior contributions cancel out, leaving only the contributions from the outer boundary CCC. This approach effectively demonstrates that the net circulation around CCC corresponds to the total flux of the vector field through DDD, confirming Green's Theorem's validity. The beauty of this proof lies in its geometric interpretation, revealing how local properties of a vector field relate to global behavior over a region.

Cayley Graph Representations

Cayley Graphs are a powerful tool used in group theory to visually represent groups and their structure. Given a group GGG and a generating set S⊆GS \subseteq GS⊆G, a Cayley graph is constructed by representing each element of the group as a vertex, and connecting vertices with directed edges based on the elements of the generating set. Specifically, there is a directed edge from vertex ggg to vertex gsgsgs for each s∈Ss \in Ss∈S. This allows for an intuitive understanding of the relationships and operations within the group. Additionally, Cayley graphs can reveal properties such as connectivity and symmetry, making them essential in both algebraic and combinatorial contexts. They are particularly useful in analyzing finite groups and can also be applied in computer science for network design and optimization problems.

Corporate Finance Valuation

Corporate finance valuation refers to the process of determining the economic value of a business or its assets. This valuation is crucial for various financial decisions, including mergers and acquisitions, investment analysis, and financial reporting. The most common methods used in corporate finance valuation include the Discounted Cash Flow (DCF) analysis, which estimates the present value of expected future cash flows, and comparative company analysis, which evaluates a company against similar firms using valuation multiples.

In DCF analysis, the formula used is:

V0=∑t=1nCFt(1+r)tV_0 = \sum_{t=1}^{n} \frac{CF_t}{(1 + r)^t}V0​=t=1∑n​(1+r)tCFt​​

where V0V_0V0​ is the present value, CFtCF_tCFt​ represents the cash flows in each period, rrr is the discount rate, and nnn is the total number of periods. Understanding these valuation techniques helps stakeholders make informed decisions regarding the financial health and potential growth of a company.

Hessian Matrix

The Hessian Matrix is a square matrix of second-order partial derivatives of a scalar-valued function. It provides important information about the local curvature of the function and is denoted as H(f)H(f)H(f) for a function fff. Specifically, for a function f:Rn→Rf: \mathbb{R}^n \rightarrow \mathbb{R}f:Rn→R, the Hessian is defined as:

H(f)=[∂2f∂x12∂2f∂x1∂x2⋯∂2f∂x1∂xn∂2f∂x2∂x1∂2f∂x22⋯∂2f∂x2∂xn⋮⋮⋱⋮∂2f∂xn∂x1∂2f∂xn∂x2⋯∂2f∂xn2]H(f) = \begin{bmatrix} \frac{\partial^2 f}{\partial x_1^2} & \frac{\partial^2 f}{\partial x_1 \partial x_2} & \cdots & \frac{\partial^2 f}{\partial x_1 \partial x_n} \\ \frac{\partial^2 f}{\partial x_2 \partial x_1} & \frac{\partial^2 f}{\partial x_2^2} & \cdots & \frac{\partial^2 f}{\partial x_2 \partial x_n} \\ \vdots & \vdots & \ddots & \vdots \\ \frac{\partial^2 f}{\partial x_n \partial x_1} & \frac{\partial^2 f}{\partial x_n \partial x_2} & \cdots & \frac{\partial^2 f}{\partial x_n^2} \end{bmatrix} H(f)=​∂x12​∂2f​∂x2​∂x1​∂2f​⋮∂xn​∂x1​∂2f​​∂x1​∂x2​∂2f​∂x22​∂2f​⋮∂xn​∂x2​∂2f​​⋯⋯⋱⋯​∂x1​∂xn​∂2f​∂x2​∂xn​∂2f​⋮∂xn2​∂2f​​​

Helmholtz Resonance

Helmholtz Resonance is a phenomenon that occurs when a cavity resonates at a specific frequency, typically due to the vibration of air within it. It is named after the German physicist Hermann von Helmholtz, who studied sound and its properties. The basic principle involves the relationship between the volume of the cavity, the neck length, and the mass of the air inside, which together determine the resonant frequency. This frequency can be calculated using the formula:

f=c2πAV⋅Lf = \frac{c}{2\pi} \sqrt{\frac{A}{V \cdot L}}f=2πc​V⋅LA​​

where:

  • fff is the resonant frequency,
  • ccc is the speed of sound in air,
  • AAA is the cross-sectional area of the neck,
  • VVV is the volume of the cavity, and
  • LLL is the effective length of the neck.

Helmholtz resonance is commonly observed in musical instruments, such as guitar bodies or brass instruments, where it enhances sound production by amplifying specific frequencies. Understanding this concept is crucial for engineers and designers involved in acoustics and sound design.

Hermite Polynomial

Hermite polynomials are a set of orthogonal polynomials that arise in probability, combinatorics, and physics, particularly in the context of quantum mechanics and the solution of differential equations. They are defined by the recurrence relation:

Hn(x)=2xHn−1(x)−2(n−1)Hn−2(x)H_n(x) = 2xH_{n-1}(x) - 2(n-1)H_{n-2}(x)Hn​(x)=2xHn−1​(x)−2(n−1)Hn−2​(x)

with the initial conditions H0(x)=1H_0(x) = 1H0​(x)=1 and H1(x)=2xH_1(x) = 2xH1​(x)=2x. The nnn-th Hermite polynomial can also be expressed in terms of the exponential function and is given by:

Hn(x)=(−1)nex2/2dndxne−x2/2H_n(x) = (-1)^n e^{x^2/2} \frac{d^n}{dx^n} e^{-x^2/2}Hn​(x)=(−1)nex2/2dxndn​e−x2/2

These polynomials are orthogonal with respect to the weight function w(x)=e−x2w(x) = e^{-x^2}w(x)=e−x2 on the interval (−∞,∞)(- \infty, \infty)(−∞,∞), meaning that:

∫−∞∞Hm(x)Hn(x)e−x2 dx=0for m≠n\int_{-\infty}^{\infty} H_m(x) H_n(x) e^{-x^2} \, dx = 0 \quad \text{for } m \neq n∫−∞∞​Hm​(x)Hn​(x)e−x2dx=0for m=n

Hermite polynomials play a crucial role in the formulation of the quantum harmonic oscillator and in the study of Gaussian integrals, making them significant in both theoretical and applied