10
9
5
10.29563
29.0546
deg
0.874157
0.485643
10.29563
10
9
5
10.29563
29.0546
deg
0.874157
0.485643
10.29563
The Gradient Calculator computes the gradient vector of a two-variable quadratic function at any specified point. Given a function of the form $$f(x, y) = ax^2 + bxy + cy^2 + dx + ey + f$$, this tool evaluates both partial derivatives, the gradient magnitude, and the gradient direction at your chosen coordinates. Whether you are studying multivariable calculus, optimizing surfaces in engineering, or analyzing scalar fields in physics, this calculator provides instant, accurate results.
The gradient is one of the most important concepts in vector calculus and mathematical analysis. Denoted $$\nabla f$$ (read "del f" or "nabla f"), the gradient of a scalar function is a vector field that points in the direction of the greatest rate of increase of the function. Its magnitude tells you how steep that increase is. At any point on a surface, the gradient vector is perpendicular to the level curve (contour line) passing through that point, providing fundamental geometric insight into the behavior of the function.
For the quadratic function $$f(x, y) = ax^2 + bxy + cy^2 + dx + ey + f$$, the partial derivatives are computed analytically. The partial derivative with respect to $$x$$ treats $$y$$ as a constant and differentiates each term containing $$x$$. Similarly, the partial derivative with respect to $$y$$ treats $$x$$ as a constant. This yields the gradient vector $$\nabla f = \left(\frac{\partial f}{\partial x},\, \frac{\partial f}{\partial y}\right)$$, which this calculator evaluates at your specified point $$(x_0, y_0)$$.
The gradient magnitude $$|\nabla f| = \sqrt{\left(\frac{\partial f}{\partial x}\right)^2 + \left(\frac{\partial f}{\partial y}\right)^2}$$ measures the maximum rate of change of the function at the given point. A magnitude of zero indicates a critical point — a local maximum, local minimum, or saddle point — where the function momentarily stops increasing or decreasing in every direction. The gradient direction, given in degrees, tells you the angle of steepest ascent measured counterclockwise from the positive $$x$$-axis.
Gradients have wide-ranging applications across science and engineering. In machine learning, gradient descent algorithms iteratively follow the negative gradient to minimize loss functions, forming the backbone of neural network training. In physics, the gradient of a potential energy function gives the force field: $$\vec{F} = -\nabla U$$. In fluid dynamics, pressure gradients drive fluid flow. In image processing, gradients detect edges by identifying regions of rapid intensity change. In economics, gradients of utility or profit functions indicate the direction of optimal resource allocation.
Quadratic functions in two variables model a rich variety of surfaces: elliptic paraboloids (bowls), hyperbolic paraboloids (saddles), and elliptic/hyperbolic cylinders. The coefficients $$a$$, $$b$$, and $$c$$ determine the curvature, while $$d$$ and $$e$$ shift the critical point away from the origin. By computing the gradient at various points, you can map out the flow lines of the surface — the paths of steepest ascent — and locate critical points where the gradient vanishes.
Enter your six coefficients and evaluation point below to instantly obtain the function value, both partial derivatives, the gradient magnitude, and the gradient direction. This tool is ideal for checking homework, verifying analytical results, or building intuition about multivariable functions.
The Gradient Calculator evaluates a quadratic function and its partial derivatives at a specified point.
Step 1: Define the function. The general two-variable quadratic is:
$$f(x, y) = ax^2 + bxy + cy^2 + dx + ey + f$$
Step 2: Compute the partial derivative with respect to x. Differentiating term by term while treating $$y$$ as a constant:
$$\frac{\partial f}{\partial x} = 2ax + by + d$$
Step 3: Compute the partial derivative with respect to y. Differentiating term by term while treating $$x$$ as a constant:
$$\frac{\partial f}{\partial y} = bx + 2cy + e$$
Step 4: Evaluate at the point. Substitute $$(x_0, y_0)$$ into both partial derivatives to obtain the gradient vector $$\nabla f(x_0, y_0)$$.
Step 5: Gradient magnitude. The magnitude is the Euclidean norm:
$$|\nabla f| = \sqrt{\left(\frac{\partial f}{\partial x}\right)^2 + \left(\frac{\partial f}{\partial y}\right)^2}$$
Step 6: Gradient direction. The angle of the gradient vector is:
$$\theta = \arctan\left(\frac{\partial f / \partial y}{\partial f / \partial x}\right)$$
converted to degrees. This gives the direction of steepest ascent at the evaluation point.
The f(x₀, y₀) value is the height of the surface at your evaluation point. This tells you where on the surface you are standing.
The ∂f/∂x value is the rate of change of the function in the x-direction at the given point. A positive value means the function increases as x increases (with y held fixed); a negative value means it decreases.
The ∂f/∂y value is the rate of change in the y-direction. Together with ∂f/∂x, these two values form the gradient vector.
The Gradient Magnitude measures the steepness of the surface — the maximum rate of change in any direction. A value of zero indicates a critical point (local extremum or saddle point).
The Gradient Direction in degrees indicates the compass heading of steepest ascent. For example, 0° means the steepest increase is in the positive x-direction; 90° means it is in the positive y-direction.
Inputs
Results
f(1,2) = 1 + 4 = 5. ∂f/∂x = 2x = 2. ∂f/∂y = 2y = 4. |∇f| = √(4+16) = √20 ≈ 4.4721. Direction = arctan(4/2) ≈ 63.43°. The gradient points away from the minimum at the origin.
Inputs
Results
f(1,2) = 1 + 4 + 4 + 3 − 2 + 0 = 10. Wait: 1 + 2(1)(2) + 4 + 3 − 2 = 1 + 4 + 4 + 3 − 2 = 10. ∂f/∂x = 2(1) + 2(2) + 3 = 9. ∂f/∂y = 2(1) + 2(2)(1) − 1 = wait: ∂f/∂y = bx + 2cy + e = 2(1) + 2(1)(2) + (−1) = 2 + 4 − 1 = 5. |∇f| = √(81+25) ≈ 10.296.
The gradient of a function $$f(x, y)$$ is the vector of its partial derivatives: $$\nabla f = (\partial f/\partial x,\, \partial f/\partial y)$$. It points in the direction of the steepest increase of the function, and its magnitude equals the rate of that increase. The gradient generalizes the concept of a derivative to functions of multiple variables.
A gradient magnitude of zero means you are at a critical point where the function has no directional preference for increase or decrease. This could be a local minimum, local maximum, or saddle point. To classify which type, you would examine the second-order partial derivatives (the Hessian matrix).
Along a level curve, the function value is constant, so the directional derivative along the curve is zero. The gradient achieves the maximum directional derivative, so it must be orthogonal to any direction of zero change. This perpendicularity is a fundamental property used in contour mapping and constrained optimization.
This calculator is specifically designed for two-variable quadratic functions of the form $$ax^2 + bxy + cy^2 + dx + ey + f$$. For higher-degree polynomials or transcendental functions, you would need a more general symbolic differentiation tool. However, quadratics are sufficient for many applications including linear regression surfaces and second-order Taylor approximations.
In machine learning, gradient descent minimizes a loss function by repeatedly moving in the direction opposite to the gradient: $$\theta_{n+1} = \theta_n - \alpha \nabla L(\theta_n)$$, where $$\alpha$$ is the learning rate. The gradient tells the algorithm which direction increases the loss the fastest, so moving opposite reduces it. This principle underlies training of neural networks, logistic regression, and many other models.
The directional derivative of $$f$$ in the direction of a unit vector $$\hat{u}$$ is $$D_{\hat{u}}f = \nabla f \cdot \hat{u}$$. This dot product is maximized when $$\hat{u}$$ points in the same direction as $$\nabla f$$, giving a maximum value equal to $$|\nabla f|$$. In the opposite direction, the directional derivative is $$-|\nabla f|$$, and perpendicular to the gradient, it is zero.
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The Roboculator Team explains calculations, planning tools, and practical formulas in clear language for real-life situations.
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