Key Insights
The North American dietary supplement market, valued at $42.31 billion in 2025, is projected to experience robust growth, exhibiting a Compound Annual Growth Rate (CAGR) of 6.10% from 2025 to 2033. This expansion is fueled by several key factors. Increasing health consciousness among consumers, coupled with a rising prevalence of chronic diseases like obesity and cardiovascular issues, is driving demand for supplements to support overall well-being and prevent health problems. The market is further boosted by the growing popularity of personalized nutrition and functional foods, with consumers actively seeking targeted supplements to address specific health needs. Furthermore, increased accessibility through various distribution channels, including online retail stores and pharmacies, contributes significantly to market growth. The diverse product portfolio, encompassing vitamins and minerals, proteins, herbal supplements, and others, caters to a broad range of consumer preferences and health goals. However, stringent regulatory frameworks and concerns regarding product safety and efficacy could pose challenges to market expansion.
Within the North American region, the United States is expected to remain the dominant market, followed by Canada and Mexico. The market segmentation reveals the popularity of various forms like tablets and capsules, and distribution channels including supermarkets/hypermarkets and pharmacies/drug stores are crucial for reaching a wide consumer base. Key players such as The Carlyle Group, Nutraceutics Inc., Procter & Gamble, Amway Corporation, and others are intensely competing, leading to product innovation and strategic partnerships. The forecast period anticipates continued growth driven by increased consumer spending on health and wellness, coupled with technological advancements in supplement formulation and delivery systems. Specific product types within the market, such as probiotics and collagen supplements, are exhibiting particularly strong growth trajectories, reflecting evolving consumer preferences.

North America Dietary Supplement Market: A Comprehensive Report (2019-2033)
This insightful report provides a detailed analysis of the North America dietary supplement market, offering crucial insights for stakeholders seeking to navigate this dynamic sector. Spanning the historical period (2019-2024), base year (2025), and forecast period (2025-2033), this comprehensive study unveils market trends, competitive landscapes, and future growth potential, valued at xx Million in 2025 and projected to reach xx Million by 2033.
North America Dietary Supplement Market Composition & Trends
This section delves into the intricate structure of the North America dietary supplement market, analyzing its concentration, innovation drivers, regulatory frameworks, substitute products, end-user demographics, and mergers & acquisitions (M&A) activities. Market share distribution amongst key players like The Carlyle Group, Nutraceutics Inc, Procter & Gamble, Amway Corporation, Bayer AG, Glanbia PLC, Forest Remedies, Herbalife Nutrition, Abbott, and NOW Food is meticulously examined. We also quantify the impact of M&A activities, revealing deal values and their influence on market consolidation.
- Market Concentration: The market exhibits a [High/Medium/Low - choose one based on data] level of concentration, with the top 5 players holding an estimated xx% market share in 2025.
- Innovation Catalysts: Technological advancements in formulation, delivery systems (e.g., targeted delivery), and personalized nutrition are driving significant innovation.
- Regulatory Landscape: The report analyzes the influence of FDA regulations and their impact on product development, marketing claims, and market access.
- Substitute Products: We explore the competitive pressure from functional foods and other alternative health products.
- End-User Profiles: A detailed segmentation of consumers based on age, lifestyle, health concerns, and purchasing behaviors is presented.
- M&A Activity: Analysis of recent M&A deals, including deal values and strategic implications for market players, is included.

North America Dietary Supplement Market Industry Evolution
This section provides a comprehensive analysis of the North America dietary supplement market's evolutionary path, tracing its growth trajectory, technological advancements, and shifts in consumer preferences from 2019 to 2033. The analysis will encompass factors such as changing health awareness, rising disposable incomes, and the increasing adoption of online retail channels. We provide specific data points illustrating growth rates (e.g., CAGR), adoption rates of new technologies, and changes in consumer demand for specific product types and forms. The impact of increasing awareness of preventative health and wellness will be thoroughly discussed.
Leading Regions, Countries, or Segments in North America Dietary Supplement Market
This section identifies the dominant regions, countries, and segments within the North America dietary supplement market. The analysis will focus on Product Type (Vitamins and Minerals, Proteins and Amino Acids, Fatty Acids, Herbal Supplements, Enzymes, Other Product Types), Form (Tablets, Capsules, Powder, Gummies, Other Forms), and Distribution Channel (Supermarkets/hypermarkets, Pharmacies/drug Stores, Online Retail Stores, Other Distribution Channels).
Key Drivers (Examples):
- Vitamins and Minerals: High consumer demand driven by increasing awareness of nutritional deficiencies.
- Online Retail Stores: Rapid growth fueled by e-commerce expansion and convenient access.
- United States: Largest market share due to high consumer spending and established distribution networks.
Dominance Factors: In-depth analysis will explore the specific factors contributing to the dominance of leading regions, countries, and segments, including investment trends, regulatory support, and consumer preferences.
North America Dietary Supplement Market Product Innovations
This section highlights recent product innovations, their applications, and performance metrics. We showcase unique selling propositions (USPs) and technological advancements that are shaping the competitive landscape. Specific examples of new formulations, delivery systems, and functional ingredients will be provided.
Propelling Factors for North America Dietary Supplement Market Growth
The growth of the North America dietary supplement market is driven by several key factors. These include increasing consumer awareness of health and wellness, rising disposable incomes enabling higher spending on health and wellness products, the expanding popularity of personalized nutrition, and supportive regulatory environments encouraging innovation. Technological advancements in formulation, delivery, and testing are also significant contributors.
Obstacles in the North America Dietary Supplement Market
The North America dietary supplement market faces several challenges. These include stringent regulatory requirements impacting product development and marketing, supply chain disruptions affecting ingredient sourcing and manufacturing, and intense competition from both established and emerging players, resulting in price pressure. These obstacles can limit market growth and profitability.
Future Opportunities in North America Dietary Supplement Market
Future opportunities include the expansion into niche segments (e.g., sports nutrition, specialized health conditions), the integration of personalized nutrition technologies, the development of innovative product formulations, and tapping into the growing demand for sustainable and ethically sourced ingredients.
Major Players in the North America Dietary Supplement Market Ecosystem
- The Carlyle Group
- Nutraceutics Inc
- Procter & Gamble
- Amway Corporation
- Bayer AG
- Glanbia PLC
- Forest Remedies
- Herbalife Nutrition
- Abbott
- NOW Food
Key Developments in North America Dietary Supplement Market Industry
- January 2024: Amway’s Nutrilite brand focuses on regenerative agriculture to ensure botanical supply, partnering with farmers.
- January 2024: Chobani launches Chobani Creations Greek Yogurt, highlighting protein, calcium, probiotics, and amino acids.
- February 2024: Medella Springs Healthcare launches AddiVance, a stimulant-free dietary supplement targeting focus and impulsivity.
Strategic North America Dietary Supplement Market Forecast
The North America dietary supplement market is poised for robust growth, driven by factors such as increasing health consciousness, rising disposable incomes, and the expansion of e-commerce. Future opportunities lie in personalized nutrition, innovative product formulations, and sustainable sourcing. The market's continued evolution promises exciting prospects for both established and new market entrants.
North America Dietary Supplement Market Segmentation
-
1. Product Type
- 1.1. Vitamins and Minerals
- 1.2. Proteins and Amino Acids
- 1.3. Fatty Acids
- 1.4. Herbal Supplements
- 1.5. Enzymes
- 1.6. Other Product Types
-
2. Form
- 2.1. Tablets
- 2.2. Capsules
- 2.3. Powder
- 2.4. Gummies
- 2.5. Other Forms
-
3. Distribution Channel
- 3.1. Supermarkets/hypermarkets
- 3.2. Pharmacies/drug Stores
- 3.3. Online Retail Stores
- 3.4. Other Distribution Channels
-
4. Geography
- 4.1. United States
- 4.2. Canada
- 4.3. Mexico
- 4.4. Rest of North America
North America Dietary Supplement Market Segmentation By Geography
- 1. United States
- 2. Canada
- 3. Mexico
- 4. Rest of North America

North America Dietary Supplement Market REPORT HIGHLIGHTS
Aspects | Details |
---|---|
Study Period | 2019-2033 |
Base Year | 2024 |
Estimated Year | 2025 |
Forecast Period | 2025-2033 |
Historical Period | 2019-2024 |
Growth Rate | CAGR of 6.10% from 2019-2033 |
Segmentation |
|
Table of Contents
- 1. Introduction
- 1.1. Research Scope
- 1.2. Market Segmentation
- 1.3. Research Methodology
- 1.4. Definitions and Assumptions
- 2. Executive Summary
- 2.1. Introduction
- 3. Market Dynamics
- 3.1. Introduction
- 3.2. Market Drivers
- 3.2.1. Increasing Focus on Maintaining Health and Well-Being; Launching Supplements For Specific Purposes and Targeted Population
- 3.3. Market Restrains
- 3.3.1. Supplement Consumption and Their Side-effects; Inclination Towards Substitute Products
- 3.4. Market Trends
- 3.4.1. Increasing Focus on Maintaining Health and Well-Being
- 4. Market Factor Analysis
- 4.1. Porters Five Forces
- 4.2. Supply/Value Chain
- 4.3. PESTEL analysis
- 4.4. Market Entropy
- 4.5. Patent/Trademark Analysis
- 5. North America Dietary Supplement Market Analysis, Insights and Forecast, 2019-2031
- 5.1. Market Analysis, Insights and Forecast - by Product Type
- 5.1.1. Vitamins and Minerals
- 5.1.2. Proteins and Amino Acids
- 5.1.3. Fatty Acids
- 5.1.4. Herbal Supplements
- 5.1.5. Enzymes
- 5.1.6. Other Product Types
- 5.2. Market Analysis, Insights and Forecast - by Form
- 5.2.1. Tablets
- 5.2.2. Capsules
- 5.2.3. Powder
- 5.2.4. Gummies
- 5.2.5. Other Forms
- 5.3. Market Analysis, Insights and Forecast - by Distribution Channel
- 5.3.1. Supermarkets/hypermarkets
- 5.3.2. Pharmacies/drug Stores
- 5.3.3. Online Retail Stores
- 5.3.4. Other Distribution Channels
- 5.4. Market Analysis, Insights and Forecast - by Geography
- 5.4.1. United States
- 5.4.2. Canada
- 5.4.3. Mexico
- 5.4.4. Rest of North America
- 5.5. Market Analysis, Insights and Forecast - by Region
- 5.5.1. United States
- 5.5.2. Canada
- 5.5.3. Mexico
- 5.5.4. Rest of North America
- 5.1. Market Analysis, Insights and Forecast - by Product Type
- 6. United States North America Dietary Supplement Market Analysis, Insights and Forecast, 2019-2031
- 6.1. Market Analysis, Insights and Forecast - by Product Type
- 6.1.1. Vitamins and Minerals
- 6.1.2. Proteins and Amino Acids
- 6.1.3. Fatty Acids
- 6.1.4. Herbal Supplements
- 6.1.5. Enzymes
- 6.1.6. Other Product Types
- 6.2. Market Analysis, Insights and Forecast - by Form
- 6.2.1. Tablets
- 6.2.2. Capsules
- 6.2.3. Powder
- 6.2.4. Gummies
- 6.2.5. Other Forms
- 6.3. Market Analysis, Insights and Forecast - by Distribution Channel
- 6.3.1. Supermarkets/hypermarkets
- 6.3.2. Pharmacies/drug Stores
- 6.3.3. Online Retail Stores
- 6.3.4. Other Distribution Channels
- 6.4. Market Analysis, Insights and Forecast - by Geography
- 6.4.1. United States
- 6.4.2. Canada
- 6.4.3. Mexico
- 6.4.4. Rest of North America
- 6.1. Market Analysis, Insights and Forecast - by Product Type
- 7. Canada North America Dietary Supplement Market Analysis, Insights and Forecast, 2019-2031
- 7.1. Market Analysis, Insights and Forecast - by Product Type
- 7.1.1. Vitamins and Minerals
- 7.1.2. Proteins and Amino Acids
- 7.1.3. Fatty Acids
- 7.1.4. Herbal Supplements
- 7.1.5. Enzymes
- 7.1.6. Other Product Types
- 7.2. Market Analysis, Insights and Forecast - by Form
- 7.2.1. Tablets
- 7.2.2. Capsules
- 7.2.3. Powder
- 7.2.4. Gummies
- 7.2.5. Other Forms
- 7.3. Market Analysis, Insights and Forecast - by Distribution Channel
- 7.3.1. Supermarkets/hypermarkets
- 7.3.2. Pharmacies/drug Stores
- 7.3.3. Online Retail Stores
- 7.3.4. Other Distribution Channels
- 7.4. Market Analysis, Insights and Forecast - by Geography
- 7.4.1. United States
- 7.4.2. Canada
- 7.4.3. Mexico
- 7.4.4. Rest of North America
- 7.1. Market Analysis, Insights and Forecast - by Product Type
- 8. Mexico North America Dietary Supplement Market Analysis, Insights and Forecast, 2019-2031
- 8.1. Market Analysis, Insights and Forecast - by Product Type
- 8.1.1. Vitamins and Minerals
- 8.1.2. Proteins and Amino Acids
- 8.1.3. Fatty Acids
- 8.1.4. Herbal Supplements
- 8.1.5. Enzymes
- 8.1.6. Other Product Types
- 8.2. Market Analysis, Insights and Forecast - by Form
- 8.2.1. Tablets
- 8.2.2. Capsules
- 8.2.3. Powder
- 8.2.4. Gummies
- 8.2.5. Other Forms
- 8.3. Market Analysis, Insights and Forecast - by Distribution Channel
- 8.3.1. Supermarkets/hypermarkets
- 8.3.2. Pharmacies/drug Stores
- 8.3.3. Online Retail Stores
- 8.3.4. Other Distribution Channels
- 8.4. Market Analysis, Insights and Forecast - by Geography
- 8.4.1. United States
- 8.4.2. Canada
- 8.4.3. Mexico
- 8.4.4. Rest of North America
- 8.1. Market Analysis, Insights and Forecast - by Product Type
- 9. Rest of North America North America Dietary Supplement Market Analysis, Insights and Forecast, 2019-2031
- 9.1. Market Analysis, Insights and Forecast - by Product Type
- 9.1.1. Vitamins and Minerals
- 9.1.2. Proteins and Amino Acids
- 9.1.3. Fatty Acids
- 9.1.4. Herbal Supplements
- 9.1.5. Enzymes
- 9.1.6. Other Product Types
- 9.2. Market Analysis, Insights and Forecast - by Form
- 9.2.1. Tablets
- 9.2.2. Capsules
- 9.2.3. Powder
- 9.2.4. Gummies
- 9.2.5. Other Forms
- 9.3. Market Analysis, Insights and Forecast - by Distribution Channel
- 9.3.1. Supermarkets/hypermarkets
- 9.3.2. Pharmacies/drug Stores
- 9.3.3. Online Retail Stores
- 9.3.4. Other Distribution Channels
- 9.4. Market Analysis, Insights and Forecast - by Geography
- 9.4.1. United States
- 9.4.2. Canada
- 9.4.3. Mexico
- 9.4.4. Rest of North America
- 9.1. Market Analysis, Insights and Forecast - by Product Type
- 10. United States North America Dietary Supplement Market Analysis, Insights and Forecast, 2019-2031
- 11. Canada North America Dietary Supplement Market Analysis, Insights and Forecast, 2019-2031
- 12. Mexico North America Dietary Supplement Market Analysis, Insights and Forecast, 2019-2031
- 13. Rest of North America North America Dietary Supplement Market Analysis, Insights and Forecast, 2019-2031
- 14. Competitive Analysis
- 14.1. Market Share Analysis 2024
- 14.2. Company Profiles
- 14.2.1 The Carlyle Group
- 14.2.1.1. Overview
- 14.2.1.2. Products
- 14.2.1.3. SWOT Analysis
- 14.2.1.4. Recent Developments
- 14.2.1.5. Financials (Based on Availability)
- 14.2.2 Nutraceutics Inc
- 14.2.2.1. Overview
- 14.2.2.2. Products
- 14.2.2.3. SWOT Analysis
- 14.2.2.4. Recent Developments
- 14.2.2.5. Financials (Based on Availability)
- 14.2.3 Procter & Gamble*List Not Exhaustive
- 14.2.3.1. Overview
- 14.2.3.2. Products
- 14.2.3.3. SWOT Analysis
- 14.2.3.4. Recent Developments
- 14.2.3.5. Financials (Based on Availability)
- 14.2.4 Amway Corporation
- 14.2.4.1. Overview
- 14.2.4.2. Products
- 14.2.4.3. SWOT Analysis
- 14.2.4.4. Recent Developments
- 14.2.4.5. Financials (Based on Availability)
- 14.2.5 Bayer AG
- 14.2.5.1. Overview
- 14.2.5.2. Products
- 14.2.5.3. SWOT Analysis
- 14.2.5.4. Recent Developments
- 14.2.5.5. Financials (Based on Availability)
- 14.2.6 Glanbia PLC
- 14.2.6.1. Overview
- 14.2.6.2. Products
- 14.2.6.3. SWOT Analysis
- 14.2.6.4. Recent Developments
- 14.2.6.5. Financials (Based on Availability)
- 14.2.7 Forest Remedies
- 14.2.7.1. Overview
- 14.2.7.2. Products
- 14.2.7.3. SWOT Analysis
- 14.2.7.4. Recent Developments
- 14.2.7.5. Financials (Based on Availability)
- 14.2.8 Herbalife Nutrition
- 14.2.8.1. Overview
- 14.2.8.2. Products
- 14.2.8.3. SWOT Analysis
- 14.2.8.4. Recent Developments
- 14.2.8.5. Financials (Based on Availability)
- 14.2.9 Abbott
- 14.2.9.1. Overview
- 14.2.9.2. Products
- 14.2.9.3. SWOT Analysis
- 14.2.9.4. Recent Developments
- 14.2.9.5. Financials (Based on Availability)
- 14.2.10 NOW Food
- 14.2.10.1. Overview
- 14.2.10.2. Products
- 14.2.10.3. SWOT Analysis
- 14.2.10.4. Recent Developments
- 14.2.10.5. Financials (Based on Availability)
- 14.2.1 The Carlyle Group
List of Figures
- Figure 1: North America Dietary Supplement Market Revenue Breakdown (Million, %) by Product 2024 & 2032
- Figure 2: North America Dietary Supplement Market Share (%) by Company 2024
List of Tables
- Table 1: North America Dietary Supplement Market Revenue Million Forecast, by Region 2019 & 2032
- Table 2: North America Dietary Supplement Market Revenue Million Forecast, by Product Type 2019 & 2032
- Table 3: North America Dietary Supplement Market Revenue Million Forecast, by Form 2019 & 2032
- Table 4: North America Dietary Supplement Market Revenue Million Forecast, by Distribution Channel 2019 & 2032
- Table 5: North America Dietary Supplement Market Revenue Million Forecast, by Geography 2019 & 2032
- Table 6: North America Dietary Supplement Market Revenue Million Forecast, by Region 2019 & 2032
- Table 7: North America Dietary Supplement Market Revenue Million Forecast, by Country 2019 & 2032
- Table 8: United States North America Dietary Supplement Market Revenue (Million) Forecast, by Application 2019 & 2032
- Table 9: Canada North America Dietary Supplement Market Revenue (Million) Forecast, by Application 2019 & 2032
- Table 10: Mexico North America Dietary Supplement Market Revenue (Million) Forecast, by Application 2019 & 2032
- Table 11: Rest of North America North America Dietary Supplement Market Revenue (Million) Forecast, by Application 2019 & 2032
- Table 12: North America Dietary Supplement Market Revenue Million Forecast, by Product Type 2019 & 2032
- Table 13: North America Dietary Supplement Market Revenue Million Forecast, by Form 2019 & 2032
- Table 14: North America Dietary Supplement Market Revenue Million Forecast, by Distribution Channel 2019 & 2032
- Table 15: North America Dietary Supplement Market Revenue Million Forecast, by Geography 2019 & 2032
- Table 16: North America Dietary Supplement Market Revenue Million Forecast, by Country 2019 & 2032
- Table 17: North America Dietary Supplement Market Revenue Million Forecast, by Product Type 2019 & 2032
- Table 18: North America Dietary Supplement Market Revenue Million Forecast, by Form 2019 & 2032
- Table 19: North America Dietary Supplement Market Revenue Million Forecast, by Distribution Channel 2019 & 2032
- Table 20: North America Dietary Supplement Market Revenue Million Forecast, by Geography 2019 & 2032
- Table 21: North America Dietary Supplement Market Revenue Million Forecast, by Country 2019 & 2032
- Table 22: North America Dietary Supplement Market Revenue Million Forecast, by Product Type 2019 & 2032
- Table 23: North America Dietary Supplement Market Revenue Million Forecast, by Form 2019 & 2032
- Table 24: North America Dietary Supplement Market Revenue Million Forecast, by Distribution Channel 2019 & 2032
- Table 25: North America Dietary Supplement Market Revenue Million Forecast, by Geography 2019 & 2032
- Table 26: North America Dietary Supplement Market Revenue Million Forecast, by Country 2019 & 2032
- Table 27: North America Dietary Supplement Market Revenue Million Forecast, by Product Type 2019 & 2032
- Table 28: North America Dietary Supplement Market Revenue Million Forecast, by Form 2019 & 2032
- Table 29: North America Dietary Supplement Market Revenue Million Forecast, by Distribution Channel 2019 & 2032
- Table 30: North America Dietary Supplement Market Revenue Million Forecast, by Geography 2019 & 2032
- Table 31: North America Dietary Supplement Market Revenue Million Forecast, by Country 2019 & 2032
Frequently Asked Questions
1. What is the projected Compound Annual Growth Rate (CAGR) of the North America Dietary Supplement Market?
The projected CAGR is approximately 6.10%.
2. Which companies are prominent players in the North America Dietary Supplement Market?
Key companies in the market include The Carlyle Group, Nutraceutics Inc, Procter & Gamble*List Not Exhaustive, Amway Corporation, Bayer AG, Glanbia PLC, Forest Remedies, Herbalife Nutrition, Abbott, NOW Food.
3. What are the main segments of the North America Dietary Supplement Market?
The market segments include Product Type , Form , Distribution Channel , Geography .
4. Can you provide details about the market size?
The market size is estimated to be USD 42.31 Million as of 2022.
5. What are some drivers contributing to market growth?
Increasing Focus on Maintaining Health and Well-Being; Launching Supplements For Specific Purposes and Targeted Population.
6. What are the notable trends driving market growth?
Increasing Focus on Maintaining Health and Well-Being.
7. Are there any restraints impacting market growth?
Supplement Consumption and Their Side-effects; Inclination Towards Substitute Products.
8. Can you provide examples of recent developments in the market?
February 2024: Medella Springs Healthcare launched AddiVance, a stimulant-free dietary supplement to address common nutritional imbalances in individuals experiencing a lack of focus and impulsive behavior. The company claimed that the product is designed using high-quality ingredients, sourced at cGMP facilities, tested, and manufactured at FDA-registered facilities in the United States.
9. What pricing options are available for accessing the report?
Pricing options include single-user, multi-user, and enterprise licenses priced at USD 4750, USD 5250, and USD 8750 respectively.
10. Is the market size provided in terms of value or volume?
The market size is provided in terms of value, measured in Million.
11. Are there any specific market keywords associated with the report?
Yes, the market keyword associated with the report is "North America Dietary Supplement Market," which aids in identifying and referencing the specific market segment covered.
12. How do I determine which pricing option suits my needs best?
The pricing options vary based on user requirements and access needs. Individual users may opt for single-user licenses, while businesses requiring broader access may choose multi-user or enterprise licenses for cost-effective access to the report.
13. Are there any additional resources or data provided in the North America Dietary Supplement Market report?
While the report offers comprehensive insights, it's advisable to review the specific contents or supplementary materials provided to ascertain if additional resources or data are available.
14. How can I stay updated on further developments or reports in the North America Dietary Supplement Market?
To stay informed about further developments, trends, and reports in the North America Dietary Supplement Market, consider subscribing to industry newsletters, following relevant companies and organizations, or regularly checking reputable industry news sources and publications.
Methodology
Step 1 - Identification of Relevant Samples Size from Population Database



Step 2 - Approaches for Defining Global Market Size (Value, Volume* & Price*)

Note*: In applicable scenarios
Step 3 - Data Sources
Primary Research
- Web Analytics
- Survey Reports
- Research Institute
- Latest Research Reports
- Opinion Leaders
Secondary Research
- Annual Reports
- White Paper
- Latest Press Release
- Industry Association
- Paid Database
- Investor Presentations

Step 4 - Data Triangulation
Involves using different sources of information in order to increase the validity of a study
These sources are likely to be stakeholders in a program - participants, other researchers, program staff, other community members, and so on.
Then we put all data in single framework & apply various statistical tools to find out the dynamic on the market.
During the analysis stage, feedback from the stakeholder groups would be compared to determine areas of agreement as well as areas of divergence