Solution 1)

This probably was deceptively easy and could have been very quickly solved without doing any calculations. This problem involves first the basic definition of a conservative vector field, and a useful theorem on conservative vector fields. 

1) A vector field  is conservative if there exists a scalar function  such that  is its' gradient. 

2) If a vector field is conservative, its' curl must be zero. 

 In other words, the curl of the gradient is always zero for any scalar function. 

In this problem we were given a scalar function . If we now compute the gradient, we obtain a vector field we will call   (the gradient is our vector field). Automatically we know it fits the definition of a conservative vector field because we know there is a scalar function which has  as its' gradient. That function is .  

Now we know that since the gradient is a conservative vector field, and therefore the curl must be equal to zero. 

Solution 2) 

Just for fun, let's see if it works by doing the actual calculation. 

Classify each vector field below as either conservative or non-conservative. 

This is a conservative vector field. This can be easily determined by computing the curl:

Because  the vector field  is conservative. 

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Connection to Physics! 

This vector field can also be shown to be conservative. In fact, all inverse square vector fields are conservative.

Common examples are Newton's Universal Law of Gravitation: 

 and Coulombs Law in electrostatics which gives the force exerted by on point charge  onto another point charge : 

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The easiest approach for a function defined in this way is to simply find a scalar function that has  as its' gradient. 

In general, the gradient of a vector field in terms of cylindrical coordinates  is written as: 

The vector field  in this example does not have any  or  dependency, so the first therm is the only non-zero component. Therefore, we only need to integrate the partial derivative in the  component in order to find

Now check to see if it works: 

Therefore, we have shown that there exists some scalar function  such that the gradient of  gives the vector field . Therefore,  is a conservative vector field. 

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In this case we must again conclude that vector field is conservative. All constant vector fields (vector fields for which every component is a constant) will always be conservative. To show this, start by integrating each component: 

Similarly, we can integrate the  and  components to obtain: 

Note that each expression for  is distinct despite the fact they're supposed to be the same function. The reason being is that we have no reason to anticipate that each component of the gradient will have all the information about the original function.

The differentiation with respect to  for instance to obtain the  component will "delete," all constant terms, and all terms consisting of variables being held constant. The trick is to include all unique terms from each of the three calculations and to simply add them and combine their constants of integration into one constant.  

As an exercise, you can test this to compute the gradient.  

You can also prove the vector field is conservative by showing that the curl is zero. This is obvious since each derivative in the curl will be equal to zero because all terms are constant. 

 The derivatives in the  component all vanish do due to the fact that neither partial derivative operators in the first row differentiate with respect to .

This shows that the curl is non-zero and therefore the vector field  is not conservative. 

In order to convert to cylindrical coordinates, we need to recall the conversion equations.

Now lets apply this to our problem.

To convert a point  into cylindrical corrdinates, the transformation equations are

.

Choices for  may vary depending on the situation, but the  coordinate remains the same.

When given Cartesian coordinates of the form  to cylindrical coordinates of the form , the first and third terms are the most straightforward.

Care should be taken, however, when calculating . The formula for it is as follows: 

However, it is important to be mindful of the signs of both  and , bearing in mind which quadrant the point lies; this will determine the value of :

It is something to bear in mind when making a calculation using a calculator; negative  values by convention create a negative , while negative  values lead to 

For our coordinates 

 (Bearing in mind sign convention)

When given Cartesian coordinates of the form  to cylindrical coordinates of the form , the first and third terms are the most straightforward.

Care should be taken, however, when calculating . The formula for it is as follows: 

However, it is important to be mindful of the signs of both  and , bearing in mind which quadrant the point lies; this will determine the value of :

It is something to bear in mind when making a calculation using a calculator; negative  values by convention create a negative , while negative  values lead to 

For our coordinates 

 (Bearing in mind sign convention)

When given Cartesian coordinates of the form  to cylindrical coordinates of the form , the first and third terms are the most straightforward.

Care should be taken, however, when calculating . The formula for it is as follows: 

However, it is important to be mindful of the signs of both  and , bearing in mind which quadrant the point lies; this will determine the value of :

It is something to bear in mind when making a calculation using a calculator; negative  values by convention create a negative , while negative  values lead to 

For our coordinates 

 (Bearing in mind sign convention)

When given Cartesian coordinates of the form  to cylindrical coordinates of the form , the first and third terms are the most straightforward.

Care should be taken, however, when calculating . The formula for it is as follows: 

However, it is important to be mindful of the signs of both  and , bearing in mind which quadrant the point lies; this will determine the value of :

It is something to bear in mind when making a calculation using a calculator; negative  values by convention create a negative , while negative  values lead to 

For our coordinates 

 (Bearing in mind sign convention)

When given Cartesian coordinates of the form  to cylindrical coordinates of the form , the first and third terms are the most straightforward.

Care should be taken, however, when calculating . The formula for it is as follows: 

However, it is important to be mindful of the signs of both  and , bearing in mind which quadrant the point lies; this will determine the value of :

It is something to bear in mind when making a calculation using a calculator; negative  values by convention create a negative , while negative  values lead to 

For our coordinates 

 (Bearing in mind sign convention)

When given Cartesian coordinates of the form  to cylindrical coordinates of the form , the first and third terms are the most straightforward.

Care should be taken, however, when calculating . The formula for it is as follows: 

However, it is important to be mindful of the signs of both  and , bearing in mind which quadrant the point lies; this will determine the value of :

It is something to bear in mind when making a calculation using a calculator; negative  values by convention create a negative , while negative  values lead to 

For our coordinates 

 (Bearing in mind sign convention)