Finding Potential Function Calculator

Calculate Potential Function

This calculator finds the potential function f(x,y,z) for a simplified conservative vector field F = (ax+b)i + (cy+d)j + (ez+g)k. Enter the coefficients and constants below.

Component	Expression	Integral w.r.t. its variable
Fx
Fy
Fz

Vector Field Components and their Integrals

What is a Potential Function Calculator?

A potential function calculator is a tool used to find a scalar function, f(x,y,z), whose gradient is equal to a given conservative vector field F(x,y,z). That is, if F = ∇f, then f is the potential function for F. In physics and engineering, conservative fields (like gravitational or electrostatic fields) are often represented by the gradient of a potential function, simplifying many calculations, such as work done or potential energy.

This specific potential function calculator helps find the potential function for a simplified vector field of the form F = (ax+b)i + (cy+d)j + (ez+g)k. For a vector field to have a potential function, it must be conservative, meaning its curl must be zero (∇ × F = 0). For our simplified field, this condition is always met.

Anyone studying vector calculus, physics (especially mechanics and electromagnetism), or engineering might use a potential function calculator. A common misconception is that all vector fields have a potential function; however, only conservative (irrotational) vector fields do.

Potential Function Formula and Mathematical Explanation

If a vector field F = Fx i + Fy j + Fz k is conservative, then there exists a scalar potential function f(x,y,z) such that:

• ∂f/∂x = Fx
• ∂f/∂y = Fy
• ∂f/∂z = Fz

To find f, we integrate these partial derivatives:

1. f(x,y,z) = ∫Fx dx + g(y,z) (integrating with respect to x, g(y,z) is the "constant" of integration)
2. Differentiate with respect to y: ∂f/∂y = ∂/∂y (∫Fx dx) + ∂g/∂y = Fy. From this, we find ∂g/∂y.
3. Integrate ∂g/∂y with respect to y: g(y,z) = ∫(Fy – ∂/∂y (∫Fx dx)) dy + h(z)
4. Substitute g(y,z) back and differentiate with respect to z to find h(z) from Fz.

For our simplified case, F = (ax+b)i + (cy+d)j + (ez+g)k:

• f(x,y,z) = ∫(ax+b) dx = (a/2)x² + bx + g(y,z)
• ∂f/∂y = ∂g/∂y = cy+d => g(y,z) = ∫(cy+d) dy = (c/2)y² + dy + h(z)
• So, f(x,y,z) = (a/2)x² + bx + (c/2)y² + dy + h(z)
• ∂f/∂z = h'(z) = ez+g => h(z) = ∫(ez+g) dz = (e/2)z² + gz + C

Thus, the potential function is: f(x,y,z) = (a/2)x² + bx + (c/2)y² + dy + (e/2)z² + gz + C, where C is the constant of integration determined by a known value of f at a point (x0, y0, z0).

Variables Table

Variable	Meaning	Unit	Typical Range
a, c, e	Coefficients of x, y, z in Fx, Fy, Fz respectively	Varies (e.g., N/m for force field)	-100 to 100
b, d, g	Constant terms in Fx, Fy, Fz respectively	Varies (e.g., N for force field)	-100 to 100
x, y, z	Coordinates	m (if F is force)	Varies
f(x,y,z)	Potential Function	J (if F is force)	Varies
C	Constant of Integration	J (if F is force)	Varies

Practical Examples (Real-World Use Cases)

Let's see how our potential function calculator works with examples.

Example 1: Simple Electrostatic Field

Suppose an electric field E (analogous to F) is given by E = (2x + 1)i + 2yj + 4k N/C. We want to find the electric potential V (analogous to -f, so f = -V).

Inputs for the potential function calculator (noting F=-E here for potential energy): a=-2, b=-1, c=-2, d=0, e=0, g=-4. Let's find the potential difference relative to the origin, so f(0,0,0)=0. x0=0, y0=0, z0=0, f(x0,y0,z0)=0.

The calculator would give: f(x,y,z) = -x² – x – y² – 4z + 0. So the electric potential V = -f = x² + x + y² + 4z.

Example 2: Gravitational Field near Earth's Surface (Simplified)

A simplified gravitational field near the surface along the z-axis (up) could be approximated as F = -mg k (where g is acceleration due to gravity, m is mass, but we consider force per unit mass, so F = -g k). Here Fx=0, Fy=0, Fz=-g. So a=0, b=0, c=0, d=0, e=0, g=-g (approx -9.8).

Inputs: a=0, b=0, c=0, d=0, e=0, g=-9.8. Let f(0,0,0)=0. x0=0, y0=0, z0=0, f(0,0,0)=0.

The potential function calculator gives f(x,y,z) = -9.8z + 0. This corresponds to the potential energy function mgh (here h=z, f is potential per unit mass).

How to Use This Potential Function Calculator

Using the potential function calculator is straightforward:

1. Enter Coefficients and Constants: Input the values for a, b, c, d, e, and g based on your vector field F = (ax+b)i + (cy+d)j + (ez+g)k.
2. Enter Reference Point (Optional): If you know the value of the potential function f at a specific point (x0, y0, z0), enter these coordinates and the value f(x0, y0, z0). This helps determine the constant of integration C. If you just want the form of f + C, you can leave x0, y0, z0, f(x0,y0,z0) as 0, and C will be relative to that.
3. Calculate: The calculator automatically updates the results as you type. You can also click "Calculate".
4. Read Results: The primary result shows the formula for f(x,y,z). Intermediate results show the constant C and the integrals of each component.
5. Review Table and Chart: The table summarizes the components and their integrals. The chart visualizes the potential function's behavior along the x and y axes.
6. Reset: Click "Reset" to return to default values.
7. Copy Results: Click "Copy Results" to copy the function, constant, and components to your clipboard.

The potential function calculator is designed for the specified simple form of F. For more complex fields, the curl must be checked first, and the integration process is more involved.

Key Factors That Affect Potential Function Results

Several factors influence the resulting potential function:

• Vector Field Components (Fx, Fy, Fz): The form of these components directly determines the form of the potential function f through integration. Our potential function calculator assumes a linear form for each component's variable part.
• Coefficients (a, c, e): These determine the quadratic terms (x², y², z²) in the potential function.
• Constants (b, d, g): These determine the linear terms (x, y, z) in the potential function.
• Conservative Nature of the Field: A potential function only exists if the field is conservative (curl F = 0). For our assumed form F = (ax+b)i + (cy+d)j + (ez+g)k, the curl is always zero, so it's always conservative. More general fields need checking.
• Reference Point (x0, y0, z0): The choice of the point where the potential function's value is known or set (e.g., f=0 at origin or infinity).
• Value at Reference Point f(x0, y0, z0): This value fixes the constant of integration C, making the potential function unique. Without it, the potential function is defined up to an arbitrary constant. Using the potential function calculator with a reference point gives a specific f.

Frequently Asked Questions (FAQ)

What is a conservative vector field?

A vector field F is conservative if the line integral of F between two points is independent of the path taken. This is equivalent to saying F is the gradient of some scalar function (the potential function), and also equivalent to saying the curl of F is zero (∇ × F = 0) in a simply connected domain.

Why is the curl of F = (ax+b)i + (cy+d)j + (ez+g)k always zero?

Curl F = (∂Fz/∂y – ∂Fy/∂z)i + (∂Fx/∂z – ∂Fz/∂x)j + (∂Fy/∂x – ∂Fx/∂y)k. For Fx=ax+b, Fy=cy+d, Fz=ez+g, all mixed partial derivatives like ∂Fz/∂y, ∂Fy/∂z, etc., are zero, so Curl F = 0.

What if my vector field is not of the form (ax+b)i + (cy+d)j + (ez+g)k?

This specific potential function calculator is designed for that form. If your field is more complex (e.g., Fx depends on y or z), you first need to verify if it's conservative (curl F = 0). If it is, you'd integrate step-by-step as outlined in the formula section, which is more complex.

What does the constant of integration C represent?

The potential function is unique only up to an additive constant. C represents this constant. In physical systems, often the difference in potential energy is important, so C cancels out, or C is fixed by choosing a reference point where the potential is defined (e.g., zero at infinity).

Can I use this potential function calculator for 2D fields?

Yes, for a 2D field F = (ax+b)i + (cy+d)j, simply set e=0 and g=0 in the calculator.

How does the potential function relate to work done?

If F is a force field, the work done by the field in moving a particle from point A to point B is W = f(A) – f(B), where f is the potential function (or W = -(f(B)-f(A)) if f is potential energy).

What if I don't know the value of f at any point?

You can still find the form of the potential function, but it will include an arbitrary constant "+ C". Our potential function calculator finds C if you provide f(x0,y0,z0), otherwise, it assumes f(0,0,0)=0 if no value is given or calculates C based on the input f(x0,y0,z0).

Is the chart always accurate?

The chart shows f(x,0,0) and f(0,y,0) over a limited range around x=0 and y=0. It gives an idea of how the potential changes along the x and y axes when other coordinates are zero, based on the calculated C.