Calc III Vector Fields: Finding Conservativity and Potential Function

In summary, the given vector field is conservative and its potential function is F(x,y,z) = 3x^3 + 4xz^3 + g(y,z), where g(y,z) is a function of y and z. The partial derivatives of this function do not match the components of the vector field, making it impossible to find a potential function. However, upon further examination, it is revealed that the vector field is indeed conservative and the potential function can be determined.
  • #1

Homework Statement



Show that the vector field given is conservative and find its potential function.

F(x, y, z)= (6x^2+4z^3)i +(4x^3y+4e^3z)j + (12xz^2+12ye^3z)k.



Homework Equations





The Attempt at a Solution


When I take partial derivative with respect of y for (6x^2+4z^3) and partial derivative with respect of x (4x^3y+4e^3z) they are not equal, then is not conservative. I don't need to check the rest of them , b/c if one fail then it's not concervative.

What about the potential function? I don't think i have to do anything any more in this problem.

I will appreciate any help. Thank you.
 
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  • #2
Yes, that's true. This vector field is NOT conservative and so does not have a potential function.

Notice what would happen if you tried to find a potential function:
You are looking for a function, F(x,y,z) such that
[tex]\nabla F= \frac{\partial F}{\partial x}\vec{i}+ \frac{\partial F}{\partial y}\vec{j}+ \frac{\partial F}{\partial x}\vec{k}= (6x^2+ 4z^3)\vec{i}+ (4x^3y+4e^{3z})\vec{j}+ (12xz^2+12ye^{3z})\vec{k}[/tex]

So we must have
[tex]\frac{\partial F}{\partial x}= 6x^2+ 4z^3[/tex]
[tex]\frac{\partial F}{\partial y}= 4x^3y+ 4e^{3z}[/tex]
[tex]\frac{\partial F}{\partial z}= 12xz^2+ 12ye^{3z}[/tex]

From the first of those, [itex]F(x,y,z)= 3x^3+ 4xz^3+ g(y,z)[/itex] since the "constant of integration" may be a function of y and z. Differentiating that with respect to y,
[tex]\frac{\partial F}{\partial y}= \frac{\partial g}{\partial y}= 4x^3y+ 4e^{3z}[/tex]
but that's impossible since the left side is a function of y and z only while the right side depends on x.

I have to ask: are you sure you have copied the problem correctly? The problem asks "Show that the vector field given is conservative", not "determine whether or not it is conservative". If the coefficient of [itex]\vec{i}[/itex] were [itex]6x^2y^2+ 4z^3[/itex], then it would be conservative.
 
  • #3
Thank you so much. I found a mistake, it is conservative and I found the potential function.
 

1. What is a vector field in Calc III?

A vector field in Calc III is a mathematical concept that describes a vector quantity at every point in space. It is represented by a function that assigns a vector to each point in a given region.

2. What are some real-life applications of vector fields?

Vector fields have various applications in different fields such as physics, engineering, and fluid dynamics. They are used to represent phenomena like force fields, magnetic fields, and fluid flow.

3. How are vector fields visualized in Calc III?

In Calc III, vector fields are typically visualized using vector plots or vector field diagrams. Vector plots show the direction and magnitude of vectors at different points, while vector field diagrams use arrows to represent the vectors at each point.

4. How are vector fields different from scalar fields?

While vector fields assign a vector to each point in space, scalar fields assign a scalar value (such as temperature, pressure, or density) to each point. Vector fields have both magnitude and direction, while scalar fields only have magnitude.

5. What are some techniques for evaluating vector fields in Calc III?

There are various techniques for evaluating vector fields in Calc III, such as line integrals, surface integrals, and the divergence and curl operations. These techniques allow for the calculation of important properties of vector fields, such as flux and circulation.

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