We write the directional derivative on a surface z=z(x,y) in terms of the grad operator acting on z as a dot product with a unit vector in the direction required.

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The scheme already discussed for writing down the reults from integration by parts is applied to cyclic integration by parts where the integrand contains a product of an exponential and a sin or cosine.

We solve simultaneously the equations of a plane and a cone and show that the intersections are circles, parabolas, ellipses, hyperbolas, straight lines or just the origin.

We solve a non-homogeneous 2nd order constant coefficient ODE with boundary conditions using the Laplace transform method. A check is applied to the solution. The process of solving is then repeated with a deliberate error and it is demonstrated that...

The quotient rule for differentiation is proved from first principles.

It is demonstrated how to perform differentiation using the chain rule, side-stepping the use of new function names such as u and v.

We calculate the Laplace transform of a rectified sine wave shape.

The product rule for differentiation is proved from first principles.

The chain rule for differentiation is proved from first principles. Use is made of the small difference formula, which is discussed first.

We find the Laplace transform of a triangle wave as an example of a periodic function.