Let f : R2 → R differentiable with f(a, b) = 5, ∂f/∂x (a, b) ≠ 0 and ∂f/∂y (a, b) ≠ 0. Let C be the intersection curve of the equation f(x, y) = 5 and u→ a nonzero vector tangent to C at the point (a, b). Detail whether each of the following statements is true or false: I. ∂f/du→ ≠ 0 necessarily. II. ∇f(a, b)·u→ = 0, necessarily III. If v→ is any vector and w→ = ∇f(a, b), then we necessarily have ∂f/dw→(a,b)≥ df/dv→(a,b). IV. r(t) = (a + df/dx (a,b) t, b - df/dy (a,b) t), t ∈ R, is the equation of the line normal to C at (a,b). V. If γ(t) = (x(t), y(t)), t ∈ I, where I is some open interval, is a parametrization of C and h(t) = f(γ(t)), then h'(α) = 0, where α ∈ I is such that γ(α) = (a, b).
Vector Arithmetic
Vectors are those objects which have a magnitude along with the direction. In vector arithmetic, we will see how arithmetic operators like addition and multiplication are used on any two vectors. Arithmetic in basic means dealing with numbers. Here, magnitude means the length or the size of an object. The notation used is the arrow over the head of the vector indicating its direction.
Vector Calculus
Vector calculus is an important branch of mathematics and it relates two important branches of mathematics namely vector and calculus.
Let f : R2 → R
Detail whether each of the following statements is true or false:
I. ∂f/du→ ≠ 0 necessarily.
II. ∇f(a, b)·u→ = 0, necessarily
III. If v→ is any vector and w→ = ∇f(a, b), then we necessarily have ∂f/dw→(a,b)≥ df/dv→(a,b).
IV. r(t) = (a + df/dx (a,b) t, b - df/dy (a,b) t), t ∈ R, is the equation of the line normal to C at (a,b).
V. If γ(t) = (x(t), y(t)), t ∈ I, where I is some open interval, is a parametrization of C and h(t) = f(γ(t)), then h'(α) = 0, where α ∈ I is such that γ(α) = (a, b).
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