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z = 2 -x² -y² in the first octant. Let C denote the boundary curve of S, priented counterclockwise when viewed from the positive z-axis. a) Write a parametrization of the surface in terms of the coordinates "x,y" of cartesian coordinates or the coordinates "r,0" from cylindrical coordinates. (x.y) [or r(r,0)] = (]. ID ar ar O) Find the formula for =x (alternatively the vector ar dr ar ar ar or dr (or Consider the vector field F(x,y,z)=(yz, – xz,1). c) Find the curl of F curl F(x.,y.z) = (]. J. D d) Use Stokes' theorem to write a double integral in polar coordinates which computes the work (circulation) of F along the curve C. Do not find the integral. For the integrand, enter the coefficients in front of the appropriate power or "r". The coefficients can be any number, including one or zero, or negative.

z = 2 -x² -y² in the first octant. Let C denote the boundary curve of S, priented counterclockwise when viewed from the positive z-axis. a) Write a parametrization of the surface in terms of the coordinates "x,y" of cartesian coordinates or the coordinates "r,0" from cylindrical coordinates. (x.y) [or r(r,0)] = (]. ID ar ar O) Find the formula for =x (alternatively the vector ar dr ar ar ar or dr (or Consider the vector field F(x,y,z)=(yz, – xz,1). c) Find the curl of F curl F(x.,y.z) = (]. J. D d) Use Stokes' theorem to write a double integral in polar coordinates which computes the work (circulation) of F along the curve C. Do not find the integral. For the integrand, enter the coefficients in front of the appropriate power or "r". The coefficients can be any number, including one or zero, or negative.

Elementary Geometry For College Students, 7e
Elementary Geometry For College Students, 7e
7th Edition
ISBN:9781337614085
Author:Alexander, Daniel C.; Koeberlein, Geralyn M.
Publisher:Alexander, Daniel C.; Koeberlein, Geralyn M.
Chapter7: Locus And Concurrence
Section7.2: Concurrence Of Lines
Problem 7E: Which lines or line segments or rays must be drawn or constructed in a triangle to locate its a...
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Consider the surface S consisting of the part of the paraboloid z=2 - x2 - y? in the first octant. Let C denote the boundary curve of S, oriented counterclockwise when viewed from the positive z-axis. a) Write a parametrization of the surface in terms of the coordinates "x,y" of cartesian coordinates or the coordinates "r,0" from cylindrical coordinates. r(x,y) [or r(r,0)] = ar b) Find the formula for dr dr ar (alternatively the vector ду dr xe dr ar ar ar (or dr = he Consider the vector field F(x,y,z)=(yz, – xz,1). dx c) Find the curl of F curl F(x,y,z) = (, ) d) Use Stokes' theorem to write a double integral in polar coordinates which computes the work (circulation) of F along the curve C. Do not find the integral. For the integrand, enter the coefficients in front of the appropriate power or "r". The coefficients can be any number, including one or zero, or negative. 02 r2 F•dr = r) dr do + 01 r1 01 = 02 = r1 = r2 =
Transcribed Image Text:Consider the surface S consisting of the part of the paraboloid z=2 - x2 - y? in the first octant. Let C denote the boundary curve of S, oriented counterclockwise when viewed from the positive z-axis. a) Write a parametrization of the surface in terms of the coordinates "x,y" of cartesian coordinates or the coordinates "r,0" from cylindrical coordinates. r(x,y) [or r(r,0)] = ar b) Find the formula for dr dr ar (alternatively the vector ду dr xe dr ar ar ar (or dr = he Consider the vector field F(x,y,z)=(yz, – xz,1). dx c) Find the curl of F curl F(x,y,z) = (, ) d) Use Stokes' theorem to write a double integral in polar coordinates which computes the work (circulation) of F along the curve C. Do not find the integral. For the integrand, enter the coefficients in front of the appropriate power or "r". The coefficients can be any number, including one or zero, or negative. 02 r2 F•dr = r) dr do + 01 r1 01 = 02 = r1 = r2 =
Consider the surface S consisting of the part of the paraboloid z=2 - x2 - y2 in the first octant. Let C denote the boundary curve of S, oriented counterclockwise when viewed from the positive z-axis. a) Write a parametrization of the surface in terms of the coordinates "x,y" of cartesian coordinates or the coordinates "r,0" from cylindrical coordinates. r(x,y) [or r(r,0)] = ( dr ar dr dr b) Find the formula for dx (alternatively the vector ду ar dr ar ar dr (or dx ^ ay dr Consider the vector field F(x,y,z)=(yz, – xz,1). c) Find the curl of F curl F(x,y,z) = (. ) d) Use Stokes' theorem to write a double integral in polar coordinates which computes the work (circulation) of F along the curve C. Do not find the integral. For the integrand, enter the coefficients in front of the appropriate power or "r". The coefficients can be any number, including one or zero, or negative. 02 r2 F•dr = r2 01 г1
Transcribed Image Text:Consider the surface S consisting of the part of the paraboloid z=2 - x2 - y2 in the first octant. Let C denote the boundary curve of S, oriented counterclockwise when viewed from the positive z-axis. a) Write a parametrization of the surface in terms of the coordinates "x,y" of cartesian coordinates or the coordinates "r,0" from cylindrical coordinates. r(x,y) [or r(r,0)] = ( dr ar dr dr b) Find the formula for dx (alternatively the vector ду ar dr ar ar dr (or dx ^ ay dr Consider the vector field F(x,y,z)=(yz, – xz,1). c) Find the curl of F curl F(x,y,z) = (. ) d) Use Stokes' theorem to write a double integral in polar coordinates which computes the work (circulation) of F along the curve C. Do not find the integral. For the integrand, enter the coefficients in front of the appropriate power or "r". The coefficients can be any number, including one or zero, or negative. 02 r2 F•dr = r2 01 г1
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