Uploaded February 2023 | Updated September 2026, 2 weeks ago
University of Oxford Mathematician Dr Tom Crawford explains the gradient vector (Grad) and the divergence (Div) for scalar and vector functions. Test yourself with this accompanying FREE worksheet in Maple Learn: learn.maplesoft.com/doc/ipubec5cip/trm-grad-and-div-worksheet
Sign-up for Maple Learn Premium using the code TOMROCKSMATHS for a discounted subscription. Head to getlearn.maplesoft.com for more information.
Check your working using the Maple Calculator App – available for free on Google Play and the App Store.
Android: play.google.com/store/apps/details?id=com.maplesoft.companion&hl=en
Apple: apps.apple.com/us/app/maple-companion/id1466659419
We begin with the formal definition of the gradient vector (Grad) and a visualisation of what it represents for a multivariable function. We then look at some examples with explicit calculation and 3D plots.
The Divergence (Div) of a vector function is then introduced - both as an equation and via the physical interpretation of what it represents. We calculate the divergence for several vector fields and then show where the notation 'Grad Dot F' comes from with a derivation.
Finally, the link between Grad, Div and the Laplacian is explored.
Don’t forget to check out the other videos in the ‘Oxford Calculus’ series – all links below.
Full playlist: youtube.com/playlist?list=PLMCRxGutHqflZoTY8JCm1GRzCdGXvZ3_S
Finding critical points for functions of several variables: youtu.be/Leomuu82-u8
Classifying critical points using the method of the discriminant: youtube.com/watch?v=5M_ts8Q2LEM
Partial differentiation explained: youtu.be/RVwcBGzQcT8
Second order linear differential equations: youtu.be/F54yhRB9qDI
Integrating factors explained: youtube.com/watch?v=ftqKuOfOX3E
Solving simple PDEs: youtu.be/uztjxrGY6Jw
Jacobians explained: youtube.com/watch?v=YqMelRryG8U
Separation of variables integration technique explained: youtu.be/zk41c0vs9XQ
Solving homogeneous first order differential equations: youtu.be/uqvqjbAcbL8
Taylor’s Theorem explained with examples and derivation: youtube.com/watch?v=DULzJmUHN5g
Heat Equation derivation: youtu.be/rz3cdzZXQms
Separable Solutions to PDEs: youtube.com/watch?v=hcm-CgHFbwI
How to solve the Heat Equation: youtu.be/l6spigOZCOs
Fourier Series Derivation: youtu.be/LEonZYoW6sk
Find out more about the Maple Calculator App and Maple Learn on the Maplesoft YouTube channel: youtube.com/channel/UCq2MmZQ8-kqEVAnmL2GSMsQ
Produced by Dr Tom Crawford at the University of Oxford. Tom is an Early-Career Teaching and Outreach Fellow at St Edmund Hall: seh.ox.ac.uk/people/tom-crawford
For more maths content check out Tom's website tomrocksmaths.com
You can also follow Tom on Facebook, Twitter and Instagram @tomrocksmaths.
facebook.com/tomrocksmaths
twitter.com/tomrocksmaths
instagram.com/tomrocksmaths
Get your Tom Rocks Maths merchandise here:
beautifulequations.net/collections/tom-rocks-maths
University of Oxford Mathematician Dr Tom Crawford explains the gradient vector (Grad) and the divergence (Div) for scalar and vector functions. Test yourself with this accompanying FREE worksheet in Maple Learn: learn.maplesoft.com/doc/ipubec5cip/trm-grad-and-div-worksheet
Sign-up for Maple Learn Premium using the code TOMROCKSMATHS for a discounted subscription. Head to getlearn.maplesoft.com for more information.
Check your working using the Maple Calculator App – available for free on Google Play and the App Store.
Android: play.google.com/store/apps/details?id=com.maplesoft.companion&hl=en
Apple: apps.apple.com/us/app/maple-companion/id1466659419
We begin with the formal definition of the gradient vector (Grad) and a visualisation of what it represents for a multivariable function. We then look at some examples with explicit calculation and 3D plots.
The Divergence (Div) of a vector function is then introduced - both as an equation and via the physical interpretation of what it represents. We calculate the divergence for several vector fields and then show where the notation 'Grad Dot F' comes from with a derivation.
Finally, the link between Grad, Div and the Laplacian is explored.
Don’t forget to check out the other videos in the ‘Oxford Calculus’ series – all links below.
Full playlist: youtube.com/playlist?list=PLMCRxGutHqflZoTY8JCm1GRzCdGXvZ3_S
Finding critical points for functions of several variables: youtu.be/Leomuu82-u8
Classifying critical points using the method of the discriminant: youtube.com/watch?v=5M_ts8Q2LEM
Partial differentiation explained: youtu.be/RVwcBGzQcT8
Second order linear differential equations: youtu.be/F54yhRB9qDI
Integrating factors explained: youtube.com/watch?v=ftqKuOfOX3E
Solving simple PDEs: youtu.be/uztjxrGY6Jw
Jacobians explained: youtube.com/watch?v=YqMelRryG8U
Separation of variables integration technique explained: youtu.be/zk41c0vs9XQ
Solving homogeneous first order differential equations: youtu.be/uqvqjbAcbL8
Taylor’s Theorem explained with examples and derivation: youtube.com/watch?v=DULzJmUHN5g
Heat Equation derivation: youtu.be/rz3cdzZXQms
Separable Solutions to PDEs: youtube.com/watch?v=hcm-CgHFbwI
How to solve the Heat Equation: youtu.be/l6spigOZCOs
Fourier Series Derivation: youtu.be/LEonZYoW6sk
Find out more about the Maple Calculator App and Maple Learn on the Maplesoft YouTube channel: youtube.com/channel/UCq2MmZQ8-kqEVAnmL2GSMsQ
Produced by Dr Tom Crawford at the University of Oxford. Tom is an Early-Career Teaching and Outreach Fellow at St Edmund Hall: seh.ox.ac.uk/people/tom-crawford
For more maths content check out Tom's website tomrocksmaths.com
You can also follow Tom on Facebook, Twitter and Instagram @tomrocksmaths.
facebook.com/tomrocksmaths
twitter.com/tomrocksmaths
instagram.com/tomrocksmaths
Get your Tom Rocks Maths merchandise here:
beautifulequations.net/collections/tom-rocks-maths
