- Let A ∈ ℒ. Let f≡ Ax. Verify from the definition that Df= A. What if f= y + Ax? Note the similarity with functions of a single variable.
- You have a level surface given by
The question is whether this deserves to be called a surface. Using the implicit function theorem, show that if f

= 0 and ifthen in some open subset of ℝ

^{3}, the relation f= 0 can be “solved” for z getting say z = zsuch that f= 0 . What happens if≠0 or≠0? Explain why z is a C^{1}map forin some open set. - Let x=
^{T}be a vector valued function defined for t ∈. Then Dx∈ℒ. We usually denote this simply as x^{′}. Thus, considered as a matrix, it is the 3 × 1 matrixthe T indicating that you take the transpose. Don’t worry too much about this. You can also consider this as a vector. What is the geometric significance of this vector? The answer is that this vector is tangent to the curve traced out by x

for t ∈. Explain why this is so using the definition of the derivative. You need to describe what is meant by being tangent first. By saying that the line x = a + tb is tangent to a parametric curve consisting of points traced out by xfor t ∈at the point a = xwhich is on both the line and the curve, you would want to haveWith this definition of what it means for a line to be tangent, explain why the line x

+ x^{′}u for u ∈is tangent to the curve determined by t → xat the point x. So why would you take the above as a definition of what it means to be tangent? Consider the component functions of x. What does the above limit say about the component functions and the corresponding components of b in terms of slopes of lines tangent to curves? - Let fbe a C
^{1}function f : U → ℝ where U is an open set in ℝ^{3}. The gradient vector, defined ashas fundamental geometric significance illustrated by the following picture.