Calculus 3 : Spherical Coordinates

Study concepts, example questions & explanations for Calculus 3

varsity tutors app store varsity tutors android store

Example Questions

← Previous 1 3 4 5 6 7 8 9 17 18

Example Question #1 : Spherical Coordinates

Convert the following into spherical coordinates.

Possible Answers:

Correct answer:

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?

Possible Answers:

Correct answer:

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 :

Quadrants

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 

Phirho

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?

Possible Answers:

Correct answer:

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 :

Quadrants

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 

Phirho

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?

Possible Answers:

Correct answer:

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 :

Quadrants

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 

Phirho

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?

Possible Answers:

Correct answer:

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 :

Quadrants

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 

Phirho

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?

Possible Answers:

Correct answer:

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 :

Quadrants

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 

Phirho

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?

Possible Answers:

Correct answer:

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 :

Quadrants

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 

Phirho

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?

Possible Answers:

Correct answer:

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 :

Quadrants

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 

Phirho

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?

Possible Answers:

Correct answer:

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 :

Quadrants

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 

Phirho

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?

Possible Answers:

Correct answer:

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 :

Quadrants

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 

Phirho

For our coordinates 

 

← Previous 1 3 4 5 6 7 8 9 17 18
Learning Tools by Varsity Tutors

Incompatible Browser

Please upgrade or download one of the following browsers to use Instant Tutoring: