All Calculus 3 Resources
Example Questions
Example Question #1 : Line Integrals
Write the parametric equations of the line that passes through the points and
.
First, you must find the vector that is parallel to the line.
This vector is
.
From the points we were given, this becomes
.
To form the parametric equations, we need to pick a point that lies on the line we want.
The point is used.
The vector form of the line is from the following equation
.
We then rewrite each expression in terms of the variables x, y, and z.
Example Question #2 : Line Integrals
Evaluate the line integral of the function
over the line segment
from
to
Evaluate the line integral using the function
over the line segment
from
to
Define the Parametric Equations to Represent
The points given lie on the line . Define the parameter
, then
can be written
. Therefore, the parametric equations for
are:
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The line integral of a function along the curve
with the parametric equation
and
with
is defined by:
(1)
Where is the vector derivative of the vector
, therefore
is simply the magnitude of the vector derivative.
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Write the vector :
Differentiate,
The absolute value (magnitude) of this vector is:
Write the function in terms of the parameter
:
Insert everything into Equation (1) noting that the limits of integration will be due to the fact that the parameter
varies from
to
over the line segment we are integrating over.
Example Question #3 : Line Integrals
Evaluate , where
, and
is any path that starts at
, and ends at
.
Since there isn't a specific path we need to take, we just evaluate at the end points.
Example Question #1 : Green's Theorem
Use Green's Theorem to evaluate , where
is a triangle with vertices
,
,
with positive orientation.
First we need to make sure that the conditions for Green's Theorem are met.
The conditions are met because it is positively oriented, piecewise smooth, simple, and closed under the region (see below).
In this particular case , and
, where
, and
refer to
.
We know from Green's Theorem that
So lets find the partial derivatives.
Example Question #2 : Green's Theorem
Use Green's Theorem to evaluate the line integral
over the region R, described by connecting the points , orientated clockwise.
Using Green's theorem
since the region is oriented clockwise, we would have
which gives us
Example Question #3 : Green's Theorem
Use Greens Theorem to evaluate the line integral
over the region connecting the points oriented clockwise
Using Green's theorem
Since the region is oriented clockwise
Example Question #1 : Divergence
Compute for
In order to find the divergence, we need to remember the formula.
Divergence Formula:
, where
,
, and
correspond to the components of a given vector field
.
Now lets apply this to our situation.
Example Question #2 : Divergence
Compute for
In order to find the divergence, we need to remember the formula.
Divergence Formula:
, where
,
, and
correspond to the components of a given vector field
.
Now lets apply this to our situation.
Example Question #3 : Divergence
Compute for
In order to find the divergence, we need to remember the formula.
Divergence Formula:
, where
,
, and
correspond to the components of a given vector field
.
Now lets apply this to our situation.
Example Question #1651 : Calculus 3
Find , where
In order to find the divergence, we need to remember the formula.
Divergence Formula:
, where
,
, and
correspond to the components of a given vector field
.
Now lets apply this to our situation.
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