Gradient Calculator
Calculate gradients, directional derivatives, and analyze vector fields with comprehensive step-by-step solutions. Our calculator handles multivariable functions, computes gradient magnitude, unit vectors, and provides detailed mathematical explanations for optimization and vector field analysis.
Last updated: December 15, 2024
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Use ^ for exponents, * for multiplication, sin, cos, ln, etc.
Gradient Analysis
Function:
Gradient Vector (∇f):
Numerical Evaluation at (x=1, y=1):
Solution Steps:
- Step 1: Identify the function f(x,y) = x^2 + y^2
- Step 2: Calculate partial derivatives for each variable
- Step 3: ∂f/∂x = 2x
- Step 4: ∂f/∂y = 2y
- Step 5: Evaluate at point (x=1, y=1)
- Step 6: ∇f = (x: 2, y: 2)
- Step 7: |∇f| = 2.8284
- Step 8: Unit vector = (x: 0.7071, y: 0.7071)
Quick Example Result
For function f(x,y) = x² + y² at point (1,1):
∇f = (2, 2)
How This Calculator Works
Our gradient calculator uses advanced differential calculus to compute gradient vectors, directional derivatives, and vector field properties. The process involves calculating partial derivatives, evaluating at specific points, and providing comprehensive vector analysis with step-by-step mathematical explanations.
Gradient Calculation Methods
Partial Derivatives:
∇f = (∂f/∂x, ∂f/∂y, ∂f/∂z, ...)Gradient Magnitude:
|∇f| = √[(∂f/∂x)² + (∂f/∂y)² + ...]Directional Derivative:
D_u f = ∇f · û = |∇f| cos θShows gradient direction, magnitude, and contour lines
Mathematical Foundation
The gradient of a function f(x₁, x₂, ..., xₙ) is defined as the vector of all its partial derivatives: ∇f = (∂f/∂x₁, ∂f/∂x₂, ..., ∂f/∂xₙ). This vector points in the direction of steepest ascent of the function and its magnitude represents the maximum rate of change. The gradient is fundamental to optimization, physics, and many areas of applied mathematics.
- Direction of steepest increase: The gradient vector points toward maximum function growth
- Magnitude significance: |∇f| gives the maximum rate of change at that point
- Orthogonal to level curves: Gradient is perpendicular to contour lines of constant function value
- Zero at critical points: ∇f = 0 indicates local maxima, minima, or saddle points
Sources & References
- Stewart Multivariable Calculus - Gradient Vectors and Directional DerivativesComprehensive coverage of gradient theory and applications
- MIT OpenCourseWare - Vector Calculus and OptimizationDetailed examples of gradient applications in optimization
- Khan Academy - Multivariable Calculus: Gradients and Directional DerivativesStep-by-step tutorials on gradient calculation techniques
Need help with other calculus concepts? Check out our partial derivative calculator and multivariable limit calculator.
Get Custom Calculator for Your BusinessExample Calculation
Given Function:
- Function: f(x,y) = x² + y²
- Evaluation Point: (1,1)
- Method: Partial derivatives
Calculation Steps:
- ∂f/∂x = 2x
- ∂f/∂y = 2y
- At (1,1): ∇f = (2×1, 2×1) = (2, 2)
- Magnitude: |∇f| = √(2² + 2²) = 2√2 ≈ 2.83
Result: ∇f(1,1) = (2, 2) with magnitude |∇f| = 2.8284
The gradient points in direction (2,2), indicating the steepest increase direction at point (1,1).
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