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

 $\displaystyle F(x,y,z)=x^2y\widehat{i}+3yz^2\widehat{j}+xz\widehat{k}$

What we will do is take the derivative of each vector element with respect to its variable $\displaystyle (\widehat{i}:x,\widehat{j}:y,\widehat{k}:z)$

Then sum the results together:

 $\displaystyle divF(x,y,z)=2xy+3z^2+x$

At the point $\displaystyle (2,4,5)$

$\displaystyle divF(2,4,5)=2(2)(4)+3(5)^2+2$

$\displaystyle divF(2,4,5)=93$

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

 $\displaystyle F(x,y,z)=4xyz\widehat{i}+2y^2z\widehat{j}+12x^2y\widehat{k}$

What we will do is take the derivative of each vector element with respect to its variable $\displaystyle (\widehat{i}:x,\widehat{j}:y,\widehat{k}:z)$

Then sum the results together:

$\displaystyle divF(x,y,z)=4yz+4yz+0=8yz$ 

At the point $\displaystyle (4,1,3)$

$\displaystyle divF(4,1,3)=8(1)(3)=24$

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

$\displaystyle F(x,y)=x^2e^y\widehat{i}+5ye^x\widehat{j}$ 

What we will do is take the derivative of each vector element with respect to its variable $\displaystyle (\widehat{i}:x,\widehat{j}:y,\widehat{k}:z)$

Then sum the results together:

$\displaystyle d(e^u)=e^udu$

$\displaystyle divF(x,y)=2xe^y+5e^x$

At the point $\displaystyle (-1,2)$

$\displaystyle divF(-1,2)=2(-1)e^2+5e^{-1}$

$\displaystyle divF(-1,2)=-12.94$

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

$\displaystyle F(x,y)=xcos(y)\widehat{i}+xcos(y)\widehat{j}$ 

What we will do is take the derivative of each vector element with respect to its variable $\displaystyle (\widehat{i}:x,\widehat{j}:y,\widehat{k}:z)$

$\displaystyle d(cos(u))=-sin(u)du$

Then sum the results together:

$\displaystyle divF(x,y)=cos(y)-xsin(y)$

At the point $\displaystyle (\pi,\pi)$

$\displaystyle divF(\pi,\pi)=cos(\pi)-\pi sin(\pi)$

$\displaystyle divF(\pi,\pi)=1$

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

$\displaystyle F(x,y)=x^2cos(y)\widehat{i}+x^2sin(y)\widehat{j}$ 

What we will do is take the derivative of each vector element with respect to its variable $\displaystyle (\widehat{i}:x,\widehat{j}:y,\widehat{k}:z)$

Then sum the results together:

$\displaystyle d(sin(u))=cos(u)du$

$\displaystyle divF(x,y)=2xcos(y)+x^2cos(y)$ 

At the point $\displaystyle (3,2\pi)$

$\displaystyle divF(3,2\pi)=2(3)cos(2\pi)+3^2cos(2\pi)=6+9=15$

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

$\displaystyle F(x,y)=e^xln(y)\widehat{i}+cos(x)sin(y)\widehat{j}$ 

What we will do is take the derivative of each vector element with respect to its variable $\displaystyle (\widehat{i}:x,\widehat{j}:y,\widehat{k}:z)$

Then sum the results together:

$\displaystyle d(e^u)=e^udu$

$\displaystyle d(sin(u))=cos(u)du$

$\displaystyle div F(x,y)=e^xln(y)+cos(x)cos(y)$

At the point $\displaystyle (2,0.2)$

$\displaystyle div F(2,0.2)=e^2ln(0.2)+cos(2)cos(0.2)=-12.3$

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

$\displaystyle F(x,y)=18xy^2\widehat{i}+21x^2y\widehat{j}$ 

What we will do is take the derivative of each vector element with respect to its variable $\displaystyle (\widehat{i}:x,\widehat{j}:y,\widehat{k}:z)$

Then sum the results together:

$\displaystyle divF(x,y)=18y^2+21x^2$

At the point  $\displaystyle (0.2,0.1)$

$\displaystyle divF(0.2,0.1)=18(0.1)^2+21(0.2)^2=1.02$

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

$\displaystyle F(x,y)=sin(xy^2)\widehat{i}+e^ycos(2x)\widehat{j}$

What we will do is take the derivative of each vector element with respect to its variable $\displaystyle (\widehat{i}:x,\widehat{j}:y,\widehat{k}:z)$

Then sum the results together:

Derivative of an exponential: 

$\displaystyle d[e^u]=e^udu$

Trigonometric derivative: 

$\displaystyle d[sin(u)]=cos(u)du$

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

$\displaystyle divF(x,y)=y^2cos(xy^2)+e^ycos(2x)$

At the point $\displaystyle (3,1)$

$\displaystyle divF(3,1)=1^2cos(3(1)^2)+e^1cos(2(3))=1.62$

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

$\displaystyle F(x,y)=x^y\widehat{i}+y^x\widehat{j}$

What we will do is take the derivative of each vector element with respect to its variable $\displaystyle (\widehat{i}:x,\widehat{j}:y,\widehat{k}:z)$

Then sum the results together:

$\displaystyle divF(x,y)=yx^{y-1}+xy^{x-1}$

At the point $\displaystyle (2,3)$

$\displaystyle divF(2,3)=(3)(2^{3-1})+(2)(3^{2-1})=18$

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

$\displaystyle F(x,y)=y^x\widehat{i}+x^y\widehat{j}$

What we will do is take the derivative of each vector element with respect to its variable $\displaystyle (\widehat{i}:x,\widehat{j}:y,\widehat{k}:z)$

Then sum the results together:

Derivative of an exponential: 

$\displaystyle d[a^u]=a^uduln(a)$

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

$\displaystyle divF(x,y)=y^xln(y)+x^yln(x)$

At the point $\displaystyle (2,3)$

$\displaystyle divF(2,3)=3^2ln(3)+2^3ln(2)=15.4$