# Calculus 3 : Spherical Coordinates

## Example Questions

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### Example Question #1 : Spherical Coordinates

Convert the following into spherical coordinates.

Explanation:

In order to convert to spherical coordinates , we need to remember the conversion equations.

Now lets apply this to our problem.

### Example Question #2 : Spherical Coordinates

A point in space is located, in Cartesian coordinates, at . What is the position of this point in spherical coordinates?

Explanation:

When given Cartesian coordinates of the form  to cylindrical coordinates of the form , it would be useful to calculate the  term first, as we'll derive  from it.

Next, begin calculating our angles. Care should be taken, however, when calculating them. Beginning with , 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

To calculate , we can make use of our  value. It is found as

, where

For our coordinates

### Example Question #3 : Spherical Coordinates

A point in space is located, in Cartesian coordinates, at . What is the position of this point in spherical coordinates?

Explanation:

When given Cartesian coordinates of the form  to cylindrical coordinates of the form , it would be useful to calculate the  term first, as we'll derive  from it.

Next, begin calculating our angles. Care should be taken, however, when calculating them. Beginning with , 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

To calculate , we can make use of our  value. It is found as

, where

For our coordinates

### Example Question #4 : Spherical Coordinates

A point in space is located, in Cartesian coordinates, at . What is the position of this point in spherical coordinates?

Explanation:

When given Cartesian coordinates of the form  to cylindrical coordinates of the form , it would be useful to calculate the  term first, as we'll derive  from it.

Next, begin calculating our angles. Care should be taken, however, when calculating them. Beginning with , 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

To calculate , we can make use of our  value. It is found as

, where

For our coordinates

### Example Question #5 : Spherical Coordinates

A point in space is located, in Cartesian coordinates, at . What is the position of this point in spherical coordinates?

Explanation:

When given Cartesian coordinates of the form  to cylindrical coordinates of the form , it would be useful to calculate the  term first, as we'll derive  from it.

Next, begin calculating our angles. Care should be taken, however, when calculating them. Beginning with , 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

To calculate , we can make use of our  value. It is found as

, where

For our coordinates

### Example Question #6 : Spherical Coordinates

A point in space is located, in Cartesian coordinates, at . What is the position of this point in spherical coordinates?

Explanation:

When given Cartesian coordinates of the form  to cylindrical coordinates of the form , it would be useful to calculate the  term first, as we'll derive  from it.

Next, begin calculating our angles. Care should be taken, however, when calculating them. Beginning with , 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

To calculate , we can make use of our  value. It is found as

, where

For our coordinates

### Example Question #7 : Spherical Coordinates

A point in space is located, in Cartesian coordinates, at . What is the position of this point in spherical coordinates?

Explanation:

When given Cartesian coordinates of the form  to cylindrical coordinates of the form , it would be useful to calculate the  term first, as we'll derive  from it.

Next, begin calculating our angles. Care should be taken, however, when calculating them. Beginning with , 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

To calculate , we can make use of our  value. It is found as

, where

For our coordinates

### Example Question #8 : Spherical Coordinates

A point in space is located, in Cartesian coordinates, at . What is the position of this point in spherical coordinates?

Explanation:

When given Cartesian coordinates of the form  to cylindrical coordinates of the form , it would be useful to calculate the  term first, as we'll derive  from it.

Next, begin calculating our angles. Care should be taken, however, when calculating them. Beginning with , 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

To calculate , we can make use of our  value. It is found as

, where

For our coordinates

### Example Question #9 : Spherical Coordinates

A point in space is located, in Cartesian coordinates, at . What is the position of this point in spherical coordinates?

Explanation:

When given Cartesian coordinates of the form  to cylindrical coordinates of the form , it would be useful to calculate the  term first, as we'll derive  from it.

Next, begin calculating our angles. Care should be taken, however, when calculating them. Beginning with , 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

To calculate , we can make use of our  value. It is found as

, where

For our coordinates

### Example Question #10 : Spherical Coordinates

A point in space is located, in Cartesian coordinates, at . What is the position of this point in spherical coordinates?

Explanation:

When given Cartesian coordinates of the form  to cylindrical coordinates of the form , it would be useful to calculate the  term first, as we'll derive  from it.

Next, begin calculating our angles. Care should be taken, however, when calculating them. Beginning with , 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

To calculate , we can make use of our  value. It is found as

, where

For our coordinates

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