Answer to Question #264739 in Calculus for JaytheCreator

Question

Question 1

(i) Construct the contour plot of the scalar field 𝑇: ℝ2 → ℝ such that 𝑇(𝑥, 𝑦) = 𝑥 2 − 𝑦.

(ii) Sketch the vector field 𝑉 ∶ ℝ2 → ℝ2 defined as follows 𝑉(𝑥, 𝑦) = (𝑥 + 𝑦, −𝑥). 

Answer 1

i)

A contour plot is a graphical technique for representing a 3-dimensional surface by plotting constant z slices, called contours, on a 2-dimensional format. That is, given a value for z, lines are drawn for connecting the (x,y) coordinates where that z value occurs.

ii)

$V(0,1)=(1,0)$

$V(1,0)=(1,-1)$

$V(-1,1)=(0,1)$

$V(1,-1)=(0,-1)$