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Directional derivative calculator

Gradient Calculator

Function f(x, y):

x:

y:

z:

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An online gradient calculator helps you to find the gradient of a straight line through two and three points. This gradient field calculator differentiates the given function to determine the gradient with step-by-step calculations. So, read on to know how to calculate gradient vectors using formulas and examples.

What is a Gradient?

In vector calculus, Gradient can refer to the derivative of a function. This term is most often used in complex situations where you have multiple inputs and only one output. The gradient vector stores all the partial derivative information of each variable.

The informal definition of gradient (also called slope) is as follows: It is a mathematical method of measuring the ascent or descent speed of a line. When the slope increases to the left, a line has a positive gradient. When a line slopes from left to right, its gradient is negative. The vertical line should have an indeterminate gradient. The symbol m is used for gradient. In algebra, differentiation can be used to find the gradient of a line or function.

However, an Online Slope Calculator helps to find the slope (m) or gradient between two points and in the Cartesian coordinate plane.

Gradient Notation:

The gradient of function f at point x is usually expressed as ∇f(x). It can also be called:

  • ∇f(x)
  • Grad f
  • ∂f/∂a
  • ∂_if and f_i

Gradient notations are also commonly used to indicate gradients. The gradient equation is defined as a unique vector field, and the scalar product of its vector v at each point x is the derivative of f along the direction of v.

$$(∇f(a)) . v = D_vf(x)$$

Cartesian Coordinates:

In the three-dimensional Cartesian coordinate system with a Euclidean metric, the gradient, if it exists, is given by:

$$∇f = ∂f/∂x a + ∂f/∂y b + ∂f/∂z c$$

Where a, b, c are the standard unit vectors in the directions of the x, y, and z coordinates, respectively.

Gradient Formula with Example:

  • Find any two points on the line you want to explore and find their Cartesian coordinates. Suppose we want to determine the slope of a straight line passing through points (8, 4) and (13, 19).
  • Take the coordinates of the first point and enter them into the gradient field calculator as \(a_1 and b_2\). Do the same for the second point, this time \(a_2 and b_2\).
  • The gradient calculator automatically uses the gradient formula and calculates it as (19-4)/(13-(8))=3.

However, an Online Directional Derivative Calculator finds the gradient and directional derivative of a function at a given point of a vector.

How to Calculate Gradient?

To calculate the gradient, we find two points, which are specified in Cartesian coordinates \((a_1, b_1) and (a_2, b_2)\). In a real example, we want to understand the interrelationship between them, that is, how high the surplus between them. This is defined by the gradient Formula:

gradient = rise / run

With rise \(= a_2-a_1, and run = b_2-b_1\). The rise is the ascent/descent of the second point relative to the first point, while running is the distance between them (horizontally).

Example:

Define gradient of a function \(x^2+y^3\) with points (1, 3).

Solution:

$$∇ (x^2+y^3)$$

$$(x, y) = (1, 3)$$

$$∇f = (∂f/∂x, ∂f/∂y)$$

Now, differentiate \(x^2 + y^3\) term by term:

Apply the power rule: \(x^2\) goes to 2x

The derivative of the constant \(y^3\) is zero.

The answer is:

$$∂f/∂x = 2x$$

Again, differentiate \(x^2 + y^3\) term by term:

The derivative of the constant \(x^2\) is zero.

Apply the power rule:  \(y^3 goes to 3y^2\)

The answer is:

$$∂f/∂y = 3y^2$$

Put the points:

$$∇f (1, 3) = (2, 27)$$

$$∇(x^2 + y^3) (x, y) = (2x, 3y^2)$$

Hence,

$$∇(x^2 + y^3) | (x, y) = (1, 3) = (2, 27)$$

How Gradient Calculator Works?

The gradient field calculator computes the gradient of a line by following these instructions:

Input:

  • Firstly, select the coordinates for the gradient.
  • Now, enter a function with two or three variables.
  • Then, substitute the values in different coordinate fields.
  • To see the answer and calculations, hit the calculate button.

Output:

  • The gradient calculator provides the standard input with a nabla sign and answer.
  • This gradient vector calculator displays step-by-step calculations to differentiate different terms.

FAQ:

What is the vector field gradient?

The gradient of the function is the vector field. It is obtained by applying the vector operator V to the scalar function f(x, y). This vector field is called a gradient (or conservative) vector field.

Does the vector gradient exist?

The gradient of a vector is a tensor that tells us how the vector field changes in any direction. We can express the gradient of a vector as its component matrix with respect to the vector field.

What is the gradient of the scalar function?

The gradient is a scalar function. The magnitude of the gradient is equal to the maximum rate of change of the scalar field, and its direction corresponds to the direction of the maximum change of the scalar function.

Is gradient a row or column vector?

The gradient is still a vector. It indicates the direction and magnitude of the fastest rate of change.

What is the potential gradient symbol?

This expression is an important feature of each conservative vector field F, that is, F has a corresponding potential ϕ. This means that the curvature of the vector field represented by ∇× disappears.

Conclusion:

Use this online gradient calculator to compute the gradients (slope) of a given function at different points. There’s no need to find the gradient by using hand and graph as it increases the uncertainty. Simply make use of our free calculator that does precise calculations for the gradient.

Reference:

From the source of Wikipedia: Motivation, Notation, Cartesian coordinates, Cylindrical and spherical coordinates, General coordinates, Gradient and the derivative or differential.

From the source of Khan Academy: Scalar-valued multivariable functions, two dimensions, three dimensions, Interpreting the gradient, gradient is perpendicular to contour lines.

From the source of Revision Math: Gradients and Graphs, Finding the gradient of a straight-line graph, Finding the gradient of a curve, Parallel Lines, Perpendicular Lines (HIGHER TIER).

From the source of Better Explained: Vector Calculus: Understanding the Gradient, Properties of the Gradient, direction of greatest increase, gradient perpendicular to lines.