Consider a surface that is parametrised by r(x,y)=(xyx2y) \mathbf{r}(x,y) = \left(\begin{array}{c} x \ y \ x^2 y \end{array} ight) .The surface area element is equal to

Question

Consider a surface that is parametrised by r(x,y)=(xyx2y) \mathbf{r}(x,y) = \left(\begin{array}{c} x \ y \ x^2 y \end{array}ight) .The surface area element is equal to

BlackTom AI · Accepted Answer

We begin by identifying the parametrization r(x,y) = (x, y, x^2 y). To obtain the surface area element, we compute the tangent vectors partial derivatives: r_x = ∂r/∂x = (1, 0, 2xy) and r_y = ∂r/∂y = (0, 1, x^2).

Next, we take their cross product r_x × r_y, which gives a vector normal to the surface with magnitude equal to the differential area element: r_x × r_y = |i   j   k; 1   0  2xy; 0   1  x^2| = (-2xy, -x^2, 1).

The area element is the norm of this cross product times dx dy: dS = |r_x × r_y| dx dy. Compute the norm: |(-2xy, -x^2, 1)| = sqrt{(-2xy)^2 + (-x^2)^2 + 1^2} = sqrt{4x^2y^2 + x^4 + 1}.

Therefore, the surface area element is dS = sqrt{4x^2 y^2 + x^4 + 1} \, dx \, dy. This matches the given expression exactly.

PHAS0009_24-25 3.3 Quiz 3 [0.25 hrs]

Consider a surface that is parametrised by r(x,y)=(xyx2y) \mathbf{r}(x,y) = \left(\begin{array}{c} x \\ y \\ x^2 y \end{array}\right) .The surface area element is equal to

View Explanation

Log in for full answers

Similar Questions

In a consumer society, many adults channel creativity into buying things

Economic stress and unpredictable times have resulted in a booming industry for self-help products

People born without creativity never can develop it

A product has a selling price of $20, a contribution margin ratio of 40% and fixed cost of $120,000. To make a profit of $30,000. The number of units that must be sold is: Type the number without $ and a comma. Eg: 20000

Which of the following statement regarding cost is correct:

Under the assumptions used in cost-volume-profit analysis, as the activity increases:

The CVP and break-even analysis are supported by a range of important assumptions. Which of the following is not an assumption of the CVP:

More Practical Tools for Students Powered by AI Study Helper

Homework AI Solver

Stylized AI Paper Writer

Plagiarism Checker Assistant

Citation AI Academic Writing Tool

In-Class Translation Assistant

AI Note Generator

AI Quiz Answers

Past Exam Questions from University Test Bank

Smart Practice Assistant

Adaptive Practice

Making Your Study Simpler