for a smooth function

USAGE SUMMARY

The phrase "for a smooth function" is correct and usable in written English.
It can be used in mathematical or analytical contexts when discussing properties or characteristics of functions that are continuous and differentiable. Example: "The theorem holds true for a smooth function, ensuring that the derivatives exist and are continuous."

✓ Grammatically correct

Science

Academia

Human-verified examples from authoritative sources

Exact Expressions

11 human-written examples

We find sufficient condition, in terms of residue currents, for a smooth function to belong to the ideal in C∞ (or Ck) generated by f.

Journal of Functional Analysis

In this paper, we propose a novel approach to recover normal derivatives for a smooth function based on its piecewise L2 projection.

Journal of Computational Physics

For a smooth function u on M, the gradient vector and the Finsler-Laplacian of u is defined by ∇ u : = L − 1 ( d u ), Δ u : = div ( ∇ u ).

Journal of Inequalities and Applications

Newton's method for solving (4.1) was presented in [16], where local quadratical convergence result was established for a smooth function ϕ.

Fixed Point Theory and Applications

For a smooth function (f:mathbb {R}^{n}times mathbb {R}^{n}rightarrow mathbb {R}), denote by (nabla_{x_{1}}f) and (nabla_{x_{2}}f) the gradient operator with respect to the first component and the second component, respectively.

Advances in Difference Equations

For a smooth function u and a smooth vector field V on M, we set M u : = { x ∈ M | d u ( x ) ≠ 0 } and M V : = { x ∈ M | V ( x ) ≠ 0 }.

Journal of Inequalities and Applications

Human-verified similar examples from authoritative sources

Similar Expressions

49 human-written examples

In this talk, we study higher-order accuracy of Berry-Esseen type and bootstrap approximations for a distribution of a smooth function of a sample average in high-dimensional non-asymptotic framework.

Princeton University

Moreover, the terms err ℓ and ϱ ℓ even converge with order O (h 3 / 2 ) which is optimal for the approximation of a smooth function by piecewise constants with respect to the H − 1 / 2 - norm.

Engineering Analysis with Boundary Elements

Maternal age was adjusted for as a smooth function.

BMJ Open

This interpolation scheme is a well known tool for generating a smooth function, interpolating the given Hermite boundary data and is summarized in Appendix A.

CAD Computer Aided Design

The policy maker's role in intermodal transport policies is to assure an environment for a smooth functioning market, maintain a complete and interoperable multimodal transport network and promote its optimised use to minimise environmental externalities.

European Transport Research Review

Expert writing Tips

Best practice

When using the phrase "for a smooth function", ensure the context clearly establishes the domain and codomain of the function, as well as the relevant smoothness conditions (e.g., differentiability, continuity of derivatives).

Common error

Avoid assuming that all functions are smooth. Explicitly state or verify the smoothness conditions relevant to your analysis, as many functions encountered in real-world applications may have discontinuities or singularities.

Linguistic Context

The phrase "for a smooth function" functions primarily as a prepositional phrase that introduces a condition or scope within a mathematical or analytical statement. As Ludwig AI confirms, it is a grammatical and usable phrase.

Expression frequency: Common

Frequent in

Science

70%

Academia

30%

Formal & Business

0%

Less common in

News & Media

0%

Encyclopedias

0%

Wiki

0%

Ludwig's WRAP-UP

The phrase "for a smooth function" is a common and grammatically correct prepositional phrase used in mathematical and analytical contexts. According to Ludwig AI, the phrase is accurate and appropriate for denoting conditions that apply specifically when dealing with functions characterized by smoothness. Usage is prevalent within scientific and academic domains, signaling a formal and precise register. Remember to verify smoothness conditions and avoid overgeneralization when employing this phrase. When seeking alternatives, consider phrases like "given a smooth function" or "where f is a smooth function" to maintain precision and semantic accuracy.

More alternative expressions

Phrases that express similar concepts, ordered by semantic similarity:

given a smooth function

Emphasizes the condition of the function being smooth as a prerequisite.

where f is a smooth function

Specifies that the function 'f' possesses the quality of being smooth.

if f is a smooth function

Presents the smoothness of the function 'f' as a conditional premise.

assuming a smooth function

Highlights the assumption that the function under consideration is smooth.

considering a smooth function

Focuses on the act of analyzing or working with a function known to be smooth.

when dealing with a smooth function

Indicates that the smoothness property is relevant within a specific analytical context.

in the case of a smooth function

Highlights the specific scenario where the function is characterized as smooth.

with the characteristic of a smooth function

Emphasizes smoothness as a defining feature or attribute.

as a continuously differentiable function

Replaces smoothness with a technical definition of continuous differentiability.

utilizing a function with continuous derivatives

Specifies the use of a function with the quality of being continuously differentiable.

FAQs

What does "smooth function" mean?

In mathematics, a smooth function is one that possesses derivatives of all orders. This implies the function is continuous and infinitely differentiable.

In what contexts is the phrase "for a smooth function" typically used?

This phrase is common in mathematical analysis, differential equations, and related fields when stating conditions or proving theorems that rely on the smoothness of a function. It's often used to specify that certain results hold true only when the function under consideration is smooth.

Can I say "for differentiable function" instead of "for a smooth function"?

While differentiability is a component of smoothness, it does not fully capture the concept. Smoothness requires infinite differentiability. Saying "for a differentiable function" may be appropriate in some contexts, but it's crucial to ensure that the level of differentiability meets the requirements of the specific theorem or analysis.

How do I determine if a function qualifies as a "smooth function"?

To determine if a function is smooth, you need to show that all its derivatives exist and are continuous. This can involve repeated differentiation and analysis of the resulting expressions. For many common functions (polynomials, trigonometric functions, exponentials), smoothness is well-established.