.calc-container {
max-width: 800px;
margin: 0 auto;
font-family: -apple-system, BlinkMacSystemFont, "Segoe UI", Roboto, Helvetica, Arial, sans-serif;
color: #333;
line-height: 1.6;
}
.calc-box {
background: #f8f9fa;
border: 1px solid #e9ecef;
border-radius: 8px;
padding: 30px;
margin-bottom: 40px;
box-shadow: 0 4px 6px rgba(0,0,0,0.05);
}
.calc-title {
text-align: center;
color: #2c3e50;
margin-bottom: 25px;
font-size: 24px;
font-weight: 700;
}
.calc-section-label {
font-weight: bold;
margin-bottom: 10px;
color: #495057;
border-bottom: 2px solid #dee2e6;
padding-bottom: 5px;
}
.input-group {
margin-bottom: 20px;
display: flex;
flex-wrap: wrap;
gap: 15px;
}
.input-wrapper {
flex: 1;
min-width: 140px;
}
.input-wrapper label {
display: block;
margin-bottom: 5px;
font-size: 14px;
font-weight: 600;
}
.input-wrapper input {
width: 100%;
padding: 10px;
border: 1px solid #ced4da;
border-radius: 4px;
font-size: 16px;
}
.function-display {
text-align: center;
font-family: "Courier New", Courier, monospace;
background: #fff;
padding: 10px;
border: 1px dashed #adb5bd;
border-radius: 4px;
margin-bottom: 20px;
color: #555;
}
.calc-btn {
display: block;
width: 100%;
background-color: #007bff;
color: white;
border: none;
padding: 15px;
font-size: 18px;
font-weight: bold;
border-radius: 4px;
cursor: pointer;
transition: background-color 0.2s;
}
.calc-btn:hover {
background-color: #0056b3;
}
.result-box {
margin-top: 25px;
background: white;
padding: 20px;
border-radius: 4px;
border-left: 5px solid #28a745;
display: none;
}
.result-row {
display: flex;
justify-content: space-between;
border-bottom: 1px solid #eee;
padding: 10px 0;
}
.result-row:last-child {
border-bottom: none;
}
.result-label {
font-weight: 600;
color: #555;
}
.result-value {
font-weight: 700;
color: #28a745;
font-family: "Courier New", Courier, monospace;
}
.article-content {
margin-top: 50px;
background: #fff;
padding: 20px;
}
.article-content h2 {
color: #2c3e50;
border-bottom: 2px solid #eee;
padding-bottom: 10px;
margin-top: 30px;
}
.article-content p {
margin-bottom: 15px;
}
.formula-box {
background: #f1f3f5;
padding: 15px;
border-left: 4px solid #007bff;
font-family: "Courier New", Courier, monospace;
margin: 20px 0;
overflow-x: auto;
}
@media (max-width: 600px) {
.input-group {
flex-direction: column;
}
}
function calculateGradient() {
// Get coefficients
var a = parseFloat(document.getElementById('coefA').value) || 0;
var b = parseFloat(document.getElementById('coefB').value) || 0;
var c = parseFloat(document.getElementById('coefC').value) || 0;
var d = parseFloat(document.getElementById('coefD').value) || 0;
var e = parseFloat(document.getElementById('coefE').value) || 0;
// Get point coordinates
var xVal = parseFloat(document.getElementById('pointX').value);
var yVal = parseFloat(document.getElementById('pointY').value);
// Validation
if (isNaN(xVal) || isNaN(yVal)) {
alert("Please enter valid numeric coordinates for Point X and Point Y.");
return;
}
// Function: f(x,y) = A*x^2 + B*y^2 + C*x*y + D*x + E*y
// Partial Derivative wrt x: fx = 2Ax + Cy + D
var fx = (2 * a * xVal) + (c * yVal) + d;
// Partial Derivative wrt y: fy = 2By + Cx + E
var fy = (2 * b * yVal) + (c * xVal) + e;
// Gradient Vector
var gradVector = "";
// Magnitude (Max Rate of Change) = sqrt(fx^2 + fy^2)
var magnitude = Math.sqrt(Math.pow(fx, 2) + Math.pow(fy, 2));
// Angle (Direction) using atan2(y, x) converting to degrees
var radians = Math.atan2(fy, fx);
var degrees = radians * (180 / Math.PI);
// Normalize angle to 0-360
if (degrees < 0) {
degrees += 360;
}
// Display Results
document.getElementById('resFx').innerHTML = fx.toFixed(4);
document.getElementById('resFy').innerHTML = fy.toFixed(4);
document.getElementById('resVector').innerHTML = gradVector;
document.getElementById('resMag').innerHTML = magnitude.toFixed(4);
document.getElementById('resAngle').innerHTML = degrees.toFixed(2) + "°";
document.getElementById('resultOutput').style.display = "block";
}
Understanding the Direction of Maximum Rate of Change
In multivariable calculus, determining how a function behaves as you move in different directions is a fundamental concept. The Direction of Maximum Rate of Change Calculator helps you find the specific vector direction in which a function increases most rapidly at a given point.
This concept relies heavily on the Gradient Vector, denoted as ∇f (del f). The gradient vector at any point P(x, y) always points in the direction of the greatest increase of the function.
What is the Gradient Vector?
The gradient of a function of two variables, f(x, y), is a vector field. At any specific point, the gradient is a vector whose components are the partial derivatives of f at that point.
∇f(x, y) = < ∂f/∂x, ∂f/∂y >
This vector has two crucial properties:
- Direction: It points in the direction where the function values increase fastest.
- Magnitude: The length (magnitude) of the vector, |∇f|, represents the actual value of that maximum rate of change (the slope).
Formulas Used in this Calculator
To provide accurate results without requiring complex symbolic entry, this calculator uses the standard quadratic form for multivariable functions:
f(x, y) = Ax² + By² + Cxy + Dx + Ey
To find the direction of maximum change, we calculate the partial derivatives:
- Partial with respect to x (fₓ): 2Ax + Cy + D
- Partial with respect to y (fᵧ): 2By + Cx + E
Once we have the specific values for fₓ and fᵧ at your chosen point (x, y), the magnitude is calculated using the Pythagorean theorem:
Magnitude = √(fₓ² + fᵧ²)
Example Calculation
Let's look at a practical example. Suppose we have a function representing a hill's elevation:
f(x, y) = 1x² + 2y² at the point (1, 1).
Here, the coefficients are A=1, B=2, and all others are 0.
- Calculate fₓ: 2(1)(1) = 2
- Calculate fᵧ: 2(2)(1) = 4
- Gradient Vector: < 2, 4 >
- Magnitude: √(2² + 4²) = √(4 + 16) = √20 ≈ 4.472
This means at the point (1,1), if you move in the direction of the vector <2, 4>, your elevation will increase at a rate of roughly 4.472 units per unit of distance.
Why is this important?
Finding the direction of maximum rate of change is essential in various fields:
- Physics: Determining the direction of force fields or heat flow.
- Machine Learning: Gradient Descent algorithms use this principle to minimize error functions by moving in the opposite direction of the gradient.
- Topography: Identifying the steepest ascent on a terrain map.