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

\(\displaystyle F(x,y)=(3x+5y)\widehat{i}+5xy^2\widehat{j}\) at \(\displaystyle (-1,2)\)

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

Then sum the results together:

\(\displaystyle divF(x,y)=(3)+(10xy)\rightarrow -17\)

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

\(\displaystyle F(x,y)=5y^2\widehat{i}+3x\widehat{j}\) at \(\displaystyle (3,4)\)

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

Then sum the results together:

\(\displaystyle divF(x,y)=(0)+(0)\rightarrow 0\)

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

\(\displaystyle F(x,y)=2x^3y^2\widehat{i}+x^2\widehat{j}\) at \(\displaystyle (1,3)\)

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

Then sum the results together:

\(\displaystyle divF(x,y)=(6x^2y^2)+(0)\rightarrow 54\)

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

\(\displaystyle F(x,y)=3xy\widehat{i}+2y\widehat{j}\) at \(\displaystyle (2,4)\)

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

Then sum the results together:

\(\displaystyle divF(x,y)=(3y)+(2)\rightarrow 14\)

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

\(\displaystyle F(x,y)=(x^2+xy)\widehat{i}+3x^3\widehat{j}\) at \(\displaystyle (-2,5)\)

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

Then sum the results together:

\(\displaystyle divF(x,y)=(2x+y)+(0)\rightarrow1\)

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

\(\displaystyle F(x,y)=(3x+3y)\widehat{i}+(2x+2y)\widehat{j}\) at \(\displaystyle (2,3)\)

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

Then sum the results together:

\(\displaystyle divF(x,y)=(3)+(2)\rightarrow5\)

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

\(\displaystyle F(x,y,z)=3xy^3\widehat{i}+4xz\widehat{j}+5x^3y\widehat{k}\) at \(\displaystyle (3,2,5)\)

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:

\(\displaystyle divF(x,y,z)=(3y^3)+(0)+(0)\rightarrow24\)

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

\(\displaystyle F(x,y,z)=4yz^2\widehat{i}+2xyz\widehat{j}+3xz\widehat{k}\) at \(\displaystyle (-2,0,4)\)

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:

\(\displaystyle divF(x,y,z)=(0)+(2xz)+(3x)\rightarrow-22\)

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

\(\displaystyle F(x,y,z)=2xz^2\widehat{i}+xy\widehat{j}-z^3\widehat{k}\) at \(\displaystyle (4,1,-1)\)

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:

\(\displaystyle divF(x,y,z)=(2z^2)+(x)+(3z^2)\rightarrow9\)

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

\(\displaystyle F(x,y,z)=9xyz\widehat{i}-x^2z^2\widehat{j}+3xyz\widehat{k}\) at \(\displaystyle (0,1,2)\)

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:

\(\displaystyle divF(x,y,z)=(9yz)+(0)+(3xy)\rightarrow18\)