Divergence can be viewed as a measure of the magnitude of a vector field's source or sink at a given point.

To visualize this, picture an open drain in a tub full of water; this drain may represent a 'sink,' and all of the velocities at each specific point in the tub represent the vector field. Close to the drain, the velocity will be greater than a spot farther away from the drain.

What divergence can calculate is what this velocity is at a given point. Again, the magnitude of the vector field.

We're given the function

What we will do is take the derivative of each vector element with respect to its variable 

Then sum the results together:

Trigonometric derivative: 

Note that u may represent large functions, and not just individual variables!

At the point 

To find the derivative of a term, multiply  by the exponent and then subtract one from the exponent. becomes , becomes , becomes -, and the derivative of  is  because it is a constant. Put those all together to get your answer: .

Divergence can be viewed as a measure of the magnitude of a vector field's source or sink at a given point.

To visualize this, picture an open drain in a tub full of water; this drain may represent a 'sink,' and all of the velocities at each specific point in the tub represent the vector field. Close to the drain, the velocity will be greater than a spot farther away from the drain.

What divergence can calculate is what this velocity is at a given point. Again, the magnitude of the vector field.

We're given the function

What we will do is take the derivative of each vector element with respect to its variable 

Then sum the results together:

Trigonometric derivative: 

Note that u may represent large functions, and not just individual variables!

At the point 

Divergence can be viewed as a measure of the magnitude of a vector field's source or sink at a given point.

To visualize this, picture an open drain in a tub full of water; this drain may represent a 'sink,' and all of the velocities at each specific point in the tub represent the vector field. Close to the drain, the velocity will be greater than a spot farther away from the drain.

What divergence can calculate is what this velocity is at a given point. Again, the magnitude of the vector field.

We're given the function

What we will do is take the derivative of each vector element with respect to its variable 

Then sum the results together:

Trigonometric derivative: 

Product rule: 

Note that  and  may represent large functions, and not just individual variables!

At the point 

The general form of Euler's method, when a derivative function, initial value, and step size are known, is:

To calculate the step size find the distance between the final and initial  value and divide by the number of steps to be used:

For this problem, we are told  and 

Knowing this, we may take the steps to estimate our function value at our final  value:

The general form of Euler's method, when a derivative function, initial value, and step size are known, is:

To calculate the step size find the distance between the final and initial  value and divide by the number of steps to be used:

For this problem, we are told  and 

Knowing this, we may take the steps to estimate our function value at our final  value:

The general form of Euler's method, when a derivative function, initial value, and step size are known, is:

To calculate the step size find the distance between the final and initial  value and divide by the number of steps to be used:

For this problem, we are told  and 

Knowing this, we may take the steps to estimate our function value at our final  value:

A line that is normal, that is to say perpendicular to a function at any given point will be normal to this slope of the line tangent to the function at that point.

The slope of the tangent can be found by taking the derivative of the function and evaluating the value of the derivative at a point of interest.

We'll need to make use of the following derivative rule(s):

Trigonometric derivative: 

Note that u may represent large functions, and not just individual variables!

Taking the derivative of the function  at the point 

The slope of the tangent is

Since the slope of the normal line is perpendicular, it is the negative reciprocal of this value

A line that is normal, that is to say perpendicular to a function at any given point will be normal to this slope of the line tangent to the function at that point.

The slope of the tangent can be found by taking the derivative of the function and evaluating the value of the derivative at a point of interest.

We'll need to make use of the following derivative rule(s):

Trigonometric derivative: 

Note that u may represent large functions, and not just individual variables!

Taking the derivative of the function  at the point 

The slope of the tangent is

Since the slope of the normal line is perpendicular, it is the negative reciprocal of this value

A line that is normal, that is to say perpendicular to a function at any given point will be normal to this slope of the line tangent to the function at that point.

The slope of the tangent can be found by taking the derivative of the function and evaluating the value of the derivative at a point of interest.

We'll need to make use of the following derivative rule(s):

Derivative of an exponential: 

Trigonometric derivative: 

Note that u may represent large functions, and not just individual variables!

Taking the derivative of the function  at the point 

The slope of the tangent is

Since the slope of the normal line is perpendicular, it is the negative reciprocal of this value