Gradient, Divergence, and Curl Calculator
Accurately compute gradient, divergence, and curl with our interactive calculator. Perfect for students and professionals dealing with vector calculus.
Full original guide (expanded)
Gradient, Divergence, and Curl Calculator
This calculator is designed for students and professionals who need to compute the gradient, divergence, and curl of vector fields in calculus. It provides accurate results and helps in understanding vector calculus concepts.
Enter Vector Field
Data Source and Methodology
All calculations are based strictly on the formulas and data provided from authoritative sources in vector calculus.
The Formula Explained
Gradient: Given by the vector of partial derivatives: \( \nabla f = \frac{\partial f}{\partial x} \hat{i} + \frac{\partial f}{\partial y} \hat{j} + \frac{\partial f}{\partial z} \hat{k} \)
Divergence: \( \nabla \cdot F = \frac{\partial F_1}{\partial x} + \frac{\partial F_2}{\partial y} + \frac{\partial F_3}{\partial z} \)
Curl: \( \nabla \times F = \left( \frac{\partial F_3}{\partial y} - \frac{\partial F_2}{\partial z} \right) \hat{i} + \left( \frac{\partial F_1}{\partial z} - \frac{\partial F_3}{\partial x} \right) \hat{j} + \left( \frac{\partial F_2}{\partial x} - \frac{\partial F_1}{\partial y} \right) \hat{k} \)
Glossary of Variables
- Vector Field (F): A function that assigns a vector to each point in space.
- Gradient (∇f): A vector indicating the direction and rate of fastest increase.
- Divergence (∇·F): A scalar representing the magnitude of a source or sink at a given point.
- Curl (∇×F): A vector describing the rotation of the field.
How It Works: A Step-by-Step Example
Consider the vector field \( F(x, y, z) = x^2 \hat{i} + y^2 \hat{j} + z^2 \hat{k} \). This calculator will compute the gradient, divergence, and curl of the given vector field using the above formulas.
Frequently Asked Questions (FAQ)
What is a Gradient?
The gradient is a vector that points in the direction of the greatest rate of increase of a function, and its magnitude is the rate of increase.
What is the Divergence?
Divergence measures the magnitude of a vector field's source or sink at a given point, in terms of a signed scalar.
What is the Curl?
Curl measures the rotation of a vector field. A curl of zero indicates the field is irrotational.
Can I use this calculator for any vector field?
Yes, as long as the vector field is defined in Cartesian coordinates.
What are some practical applications of these calculations?
These calculations are used in fluid dynamics, electromagnetism, and more to analyze physical phenomena.
Formula (LaTeX) + variables + units
','
Gradient: Given by the vector of partial derivatives: \( \nabla f = \frac{\partial f}{\partial x} \hat{i} + \frac{\partial f}{\partial y} \hat{j} + \frac{\partial f}{\partial z} \hat{k} \) Divergence: \( \nabla \cdot F = \frac{\partial F_1}{\partial x} + \frac{\partial F_2}{\partial y} + \frac{\partial F_3}{\partial z} \) Curl: \( \nabla \times F = \left( \frac{\partial F_3}{\partial y} - \frac{\partial F_2}{\partial z} \right) \hat{i} + \left( \frac{\partial F_1}{\partial z} - \frac{\partial F_3}{\partial x} \right) \hat{j} + \left( \frac{\partial F_2}{\partial x} - \frac{\partial F_1}{\partial y} \right) \hat{k} \)
- No variables provided in audit spec.
- NIST — Weights and measures — nist.gov · Accessed 2026-01-19
https://www.nist.gov/pml/weights-and-measures - FTC — Consumer advice — consumer.ftc.gov · Accessed 2026-01-19
https://consumer.ftc.gov/
Last code update: 2026-01-19
- Initial audit spec draft generated from HTML extraction (review required).
- Verify formulas match the calculator engine and convert any text-only formulas to LaTeX.
- Confirm sources are authoritative and relevant to the calculator methodology.
Gradient, Divergence, and Curl Calculator
This calculator is designed for students and professionals who need to compute the gradient, divergence, and curl of vector fields in calculus. It provides accurate results and helps in understanding vector calculus concepts.
Enter Vector Field
Data Source and Methodology
All calculations are based strictly on the formulas and data provided from authoritative sources in vector calculus.
The Formula Explained
Gradient: Given by the vector of partial derivatives: \( \nabla f = \frac{\partial f}{\partial x} \hat{i} + \frac{\partial f}{\partial y} \hat{j} + \frac{\partial f}{\partial z} \hat{k} \)
Divergence: \( \nabla \cdot F = \frac{\partial F_1}{\partial x} + \frac{\partial F_2}{\partial y} + \frac{\partial F_3}{\partial z} \)
Curl: \( \nabla \times F = \left( \frac{\partial F_3}{\partial y} - \frac{\partial F_2}{\partial z} \right) \hat{i} + \left( \frac{\partial F_1}{\partial z} - \frac{\partial F_3}{\partial x} \right) \hat{j} + \left( \frac{\partial F_2}{\partial x} - \frac{\partial F_1}{\partial y} \right) \hat{k} \)
Glossary of Variables
- Vector Field (F): A function that assigns a vector to each point in space.
- Gradient (∇f): A vector indicating the direction and rate of fastest increase.
- Divergence (∇·F): A scalar representing the magnitude of a source or sink at a given point.
- Curl (∇×F): A vector describing the rotation of the field.
How It Works: A Step-by-Step Example
Consider the vector field \( F(x, y, z) = x^2 \hat{i} + y^2 \hat{j} + z^2 \hat{k} \). This calculator will compute the gradient, divergence, and curl of the given vector field using the above formulas.
Frequently Asked Questions (FAQ)
What is a Gradient?
The gradient is a vector that points in the direction of the greatest rate of increase of a function, and its magnitude is the rate of increase.
What is the Divergence?
Divergence measures the magnitude of a vector field's source or sink at a given point, in terms of a signed scalar.
What is the Curl?
Curl measures the rotation of a vector field. A curl of zero indicates the field is irrotational.
Can I use this calculator for any vector field?
Yes, as long as the vector field is defined in Cartesian coordinates.
What are some practical applications of these calculations?
These calculations are used in fluid dynamics, electromagnetism, and more to analyze physical phenomena.
Formula (LaTeX) + variables + units
','
Gradient: Given by the vector of partial derivatives: \( \nabla f = \frac{\partial f}{\partial x} \hat{i} + \frac{\partial f}{\partial y} \hat{j} + \frac{\partial f}{\partial z} \hat{k} \) Divergence: \( \nabla \cdot F = \frac{\partial F_1}{\partial x} + \frac{\partial F_2}{\partial y} + \frac{\partial F_3}{\partial z} \) Curl: \( \nabla \times F = \left( \frac{\partial F_3}{\partial y} - \frac{\partial F_2}{\partial z} \right) \hat{i} + \left( \frac{\partial F_1}{\partial z} - \frac{\partial F_3}{\partial x} \right) \hat{j} + \left( \frac{\partial F_2}{\partial x} - \frac{\partial F_1}{\partial y} \right) \hat{k} \)
- No variables provided in audit spec.
- NIST — Weights and measures — nist.gov · Accessed 2026-01-19
https://www.nist.gov/pml/weights-and-measures - FTC — Consumer advice — consumer.ftc.gov · Accessed 2026-01-19
https://consumer.ftc.gov/
Last code update: 2026-01-19
- Initial audit spec draft generated from HTML extraction (review required).
- Verify formulas match the calculator engine and convert any text-only formulas to LaTeX.
- Confirm sources are authoritative and relevant to the calculator methodology.
Gradient, Divergence, and Curl Calculator
This calculator is designed for students and professionals who need to compute the gradient, divergence, and curl of vector fields in calculus. It provides accurate results and helps in understanding vector calculus concepts.
Enter Vector Field
Data Source and Methodology
All calculations are based strictly on the formulas and data provided from authoritative sources in vector calculus.
The Formula Explained
Gradient: Given by the vector of partial derivatives: \( \nabla f = \frac{\partial f}{\partial x} \hat{i} + \frac{\partial f}{\partial y} \hat{j} + \frac{\partial f}{\partial z} \hat{k} \)
Divergence: \( \nabla \cdot F = \frac{\partial F_1}{\partial x} + \frac{\partial F_2}{\partial y} + \frac{\partial F_3}{\partial z} \)
Curl: \( \nabla \times F = \left( \frac{\partial F_3}{\partial y} - \frac{\partial F_2}{\partial z} \right) \hat{i} + \left( \frac{\partial F_1}{\partial z} - \frac{\partial F_3}{\partial x} \right) \hat{j} + \left( \frac{\partial F_2}{\partial x} - \frac{\partial F_1}{\partial y} \right) \hat{k} \)
Glossary of Variables
- Vector Field (F): A function that assigns a vector to each point in space.
- Gradient (∇f): A vector indicating the direction and rate of fastest increase.
- Divergence (∇·F): A scalar representing the magnitude of a source or sink at a given point.
- Curl (∇×F): A vector describing the rotation of the field.
How It Works: A Step-by-Step Example
Consider the vector field \( F(x, y, z) = x^2 \hat{i} + y^2 \hat{j} + z^2 \hat{k} \). This calculator will compute the gradient, divergence, and curl of the given vector field using the above formulas.
Frequently Asked Questions (FAQ)
What is a Gradient?
The gradient is a vector that points in the direction of the greatest rate of increase of a function, and its magnitude is the rate of increase.
What is the Divergence?
Divergence measures the magnitude of a vector field's source or sink at a given point, in terms of a signed scalar.
What is the Curl?
Curl measures the rotation of a vector field. A curl of zero indicates the field is irrotational.
Can I use this calculator for any vector field?
Yes, as long as the vector field is defined in Cartesian coordinates.
What are some practical applications of these calculations?
These calculations are used in fluid dynamics, electromagnetism, and more to analyze physical phenomena.
Formula (LaTeX) + variables + units
','
Gradient: Given by the vector of partial derivatives: \( \nabla f = \frac{\partial f}{\partial x} \hat{i} + \frac{\partial f}{\partial y} \hat{j} + \frac{\partial f}{\partial z} \hat{k} \) Divergence: \( \nabla \cdot F = \frac{\partial F_1}{\partial x} + \frac{\partial F_2}{\partial y} + \frac{\partial F_3}{\partial z} \) Curl: \( \nabla \times F = \left( \frac{\partial F_3}{\partial y} - \frac{\partial F_2}{\partial z} \right) \hat{i} + \left( \frac{\partial F_1}{\partial z} - \frac{\partial F_3}{\partial x} \right) \hat{j} + \left( \frac{\partial F_2}{\partial x} - \frac{\partial F_1}{\partial y} \right) \hat{k} \)
- No variables provided in audit spec.
- NIST — Weights and measures — nist.gov · Accessed 2026-01-19
https://www.nist.gov/pml/weights-and-measures - FTC — Consumer advice — consumer.ftc.gov · Accessed 2026-01-19
https://consumer.ftc.gov/
Last code update: 2026-01-19
- Initial audit spec draft generated from HTML extraction (review required).
- Verify formulas match the calculator engine and convert any text-only formulas to LaTeX.
- Confirm sources are authoritative and relevant to the calculator methodology.