Oxford Calculus: Gradient (Grad) and Divergence (Div) Explained

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: https://learn.maplesoft.com/doc/ipube...

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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:    • Oxford Calculus  

Finding critical points for functions of several variables:    • Oxford Calculus: Finding Critical Poi...  

Classifying critical points using the method of the discriminant:    • Oxford Calculus: Classifying 2D Criti...  

Partial differentiation explained:    • Oxford Calculus: Partial Differentiat...  

Second order linear differential equations:    • Oxford Mathematics Open Day 2021: Dif...  

Integrating factors explained:    • Oxford Calculus: Integrating Factors ...  

Solving simple PDEs:    • Oxford Calculus: Solving Simple PDEs  

Jacobians explained:    • Oxford Calculus: Jacobians Explained  

Separation of variables integration technique explained:    • Oxford Calculus: Separation of Variab...  

Solving homogeneous first order differential equations:    • Oxford Calculus: Solving Homogeneous ...  

Taylor’s Theorem explained with examples and derivation:    • Oxford Calculus: Taylor's Theorem Exp...  

Heat Equation derivation:    • Oxford Calculus: Heat Equation Deriva...  

Separable Solutions to PDEs:    • Oxford Calculus: Separable Solutions ...  

How to solve the Heat Equation:    • Oxford Calculus: How to Solve the Hea...  

Fourier Series Derivation:    • Oxford Calculus: Fourier Series Deriv...  

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Produced by Dr Tom Crawford at the University of Oxford. Tom is an Early-Career Teaching and Outreach Fellow at St Edmund Hall: https://www.seh.ox.ac.uk/people/tom-c...

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