By Kusahara T.
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Additional info for A barrier method for quasilinear ordinary differential equations of the curvature type
To state precisely where these tools lead would require a whole book in itself, and even then would only cover the tip of the iceberg. Throughout the rest of this book you’ll be using these techniques, and so you’ll be able to gain some insight into where you might employ speciﬁc mathematical techniques. In this chapter we’ve touched on a couple of ideas that may be aided by a lengthier explanation than was already given. If you ﬁnd these sorts of proofs helpful, then we recommend you take a careful look at what follows.
Give your answer in spherical polar coordinates. 6. What is the general polar equation of the shape that looks like the inﬁnity sign? 7. What is the general polar equation of the shape that looks like a ﬁgure 8? 8. What is the general polar equation of a lima¸con? 9. What is the general polar equation of a cardioid? 10. What type of shape is deﬁned by an equation of the form r = a+bθ? Where Now? In this chapter we’ve dealt mainly with how points and shapes are expressed using polar coordinates.
Here’s an example of the rule in action, which you should hopefully be more comfortable with. If you really haven’t seen the chain rule before, you’ll deﬁnitely need to go and look it up before proceeding. Evaluate d (sin(2x)) dx The ﬁrst thing to do is to identify how this function relates to the standard form of M (N (x)). We know that M (N (x)) is sin(2x), and so N (x) must be 2x here. Therefore N (x), which is “the derivative of N with respect to x,” is equal to 2, and M (N (x)), which is “the derivative of M with respect to N (x),” is equal to cos(N (x)).
A barrier method for quasilinear ordinary differential equations of the curvature type by Kusahara T.