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Let C be a triangle in the ry-plane with vertices (ıv). (2.92), and (T3, Vs) arranged so that C' ...
Let C be a triangle in the x-y plane with vertices (x1,y1), (x2y2) and (x3,y3) arranged so that C...
Let C be a triangle in the x-y
plane with vertices (x1,y1), (x2y2) and (x3,y3) arranged so that C
is positively-oriented.
Let C be a triangle in the xy-plane with vertices (x,y), (z2,p), and (z3,U3) arranged so that C is positively-oriented. a.) Sketch such a triangle and indicate its orientation. b.) Apply Green's Theorem to compute the area of the triangle as a (sum of) path integral(s) around the boundary. Get a formula for area in terms of the coordinates...
9. [15 Points) Let C be the boundary of the triangle with vertices (1, 1), (2,...
9. [15 Points) Let C be the boundary of the triangle with vertices (1, 1), (2, 3) and (2, 1), oriented positively i.e. counterclockwise). Let F be the vector field F(1, y) = (e* + y²)i + (ry + cos y)j. Compute the line integral F. dr. 10. (15 Points) Let S be the portion of the paraboloid z = 1-rº-ythat lies on and above the plane z = 0. S is oriented by the normal directed upwards. If F...
T3 finite element is defined over AABC (in physical coordinates). The vertices of this triangle hav the follow...
T3 finite element is defined over AABC (in physical coordinates). The vertices of this triangle hav the following coordinates: A(-2,-1), B(3,2), and C(0,6). Problem 2 a) Using 1 point and 3 point integration rules, compute f(x, y)ds AABC 2x2-3xy + y2. where f(x,y) Which rule gives more accurate result? c) What is the integration error, if 3 point rule is used? (Hint: for what polynomial degree 3 point rule gives the exact result?) b)
T3 finite element is defined over...
4. Let - xy’i +3yj , and let C be the counterclockwise oriented triangle whose vertices...
4. Let - xy’i +3yj , and let C be the counterclockwise oriented triangle whose vertices are 0(0,0), P(2,0) and Q(2,8). Using Green's Theorem, ايثار a) Sf(-2x)dyex b) ſj(-2xy)dydx c) ff(xv* + 3y Mdvdx 0555(xv? +3 y Ddydx e) none of these 00 0 0
Let A be the inside and boundary of the triangle in R2 whose vertices are (0,0), (1,0) and (0,1). Let C be the curve obtained by proceeding around the boundary of A in an anti- clockwise direction. P...
Let A be the inside and boundary of the triangle in R2 whose vertices are (0,0), (1,0) and (0,1). Let C be the curve obtained by proceeding around the boundary of A in an anti- clockwise direction. Prove İ}!").lx (ly İ)(2 dr dy. Pdr+Qdy That is, prove Green's Theorem for the triangle A. [Hint: the lecture notes have a proof for when A is a rectangle. So, the idea is is to give a similar proof where we have this...
7. Evaluate (6x - 6y+8) dx+(4 +9y +7) dy where C is the boundary of the triangle in the ry plane, wit h vertices at (0,...
7. Evaluate (6x - 6y+8) dx+(4 +9y +7) dy where C is the boundary of the triangle in the ry plane, wit h vertices at (0,0), (1,0)and (1,4) traversed once anticlockwise. (a) 10 (c) 20 (b)-8 (d) 8 10. Find the flux of F =-rit 2yj otward across the ellipse-+ -1. (a) 36π (b) 18m (c) o (d) 6π
7. Evaluate (6x - 6y+8) dx+(4 +9y +7) dy where C is the boundary of the triangle in the ry plane,...
(a) Let C be the line segment on the plane that starts from a point (xi,yi) to a different point (x2,Y2). Show that (b) Consider a simple polygon whose vertices are (2.1 , Й), (T2, Уг), . . . , (Xn,...
(a) Let C be the line segment on the plane that starts from a point (xi,yi) to a different point (x2,Y2). Show that (b) Consider a simple polygon whose vertices are (2.1 , Й), (T2, Уг), . . . , (Xn, yn) if its boundary is traversed counterclockwise. Use Green's theorem to show that the area of this polygon is
(a) Let C be the line segment on the plane that starts from a point (xi,yi) to a different point...
3. (2 Points) Let Q be the quadrilateral in the ry-plane with vertices (1, 0), (4,0), (0, 1), (0,4). Consider 1 dA...
3. (2 Points) Let Q be the quadrilateral in the ry-plane with vertices (1, 0), (4,0), (0, 1), (0,4). Consider 1 dA I+y Deda (a) Evaluate the integral using the normal ry-coordinates. (b) Consider the change of coordinates r = u-uv and y= uv. What is the image of Q under this change of coordinates?bi (c) Calculate the integral using the change of coordinates from the previous part. Change of Variables When working integrals, it is wise to choose a...
need 1-5 Midterm #3, Math 228 Each question is worth five points. 1. Let F(r.yzy). Let C be any curve that goes from...
need 1-5
Midterm #3, Math 228 Each question is worth five points. 1. Let F(r.yzy). Let C be any curve that goes from A(-1,3,9) to B(1,6,-4). a) Show that F is conservative. b) Find a function φ such that ▽φ = F c) Use the result of b) to find Ic F Tds 2. Let F(z, y)-(2), and let C be the boundary of the square with vertices (1, 1). (-1,1). (-1,-1 traced out in the counter-clockwise direction. Find Jc...
Let C be a triangle in the x-y
plane with vertices (x1,y1), (x2y2) and (x3,y3) arranged so that C
is positively-oriented.
Let C be a triangle in the xy-plane with vertices (x,y), (z2,p), and (z3,U3) arranged so that C is positively-oriented. a.) Sketch such a triangle and indicate its orientation. b.) Apply Green's Theorem to compute the area of the triangle as a (sum of) path integral(s) around the boundary. Get a formula for area in terms of the coordinates...
9. [15 Points) Let C be the boundary of the triangle with vertices (1, 1), (2, 3) and (2, 1), oriented positively i.e. counterclockwise). Let F be the vector field F(1, y) = (e* + y²)i + (ry + cos y)j. Compute the line integral F. dr. 10. (15 Points) Let S be the portion of the paraboloid z = 1-rº-ythat lies on and above the plane z = 0. S is oriented by the normal directed upwards. If F...
T3 finite element is defined over AABC (in physical coordinates). The vertices of this triangle hav the following coordinates: A(-2,-1), B(3,2), and C(0,6). Problem 2 a) Using 1 point and 3 point integration rules, compute f(x, y)ds AABC 2x2-3xy + y2. where f(x,y) Which rule gives more accurate result? c) What is the integration error, if 3 point rule is used? (Hint: for what polynomial degree 3 point rule gives the exact result?) b)
T3 finite element is defined over...
4. Let - xy’i +3yj , and let C be the counterclockwise oriented triangle whose vertices are 0(0,0), P(2,0) and Q(2,8). Using Green's Theorem, ايثار a) Sf(-2x)dyex b) ſj(-2xy)dydx c) ff(xv* + 3y Mdvdx 0555(xv? +3 y Ddydx e) none of these 00 0 0
Let A be the inside and boundary of the triangle in R2 whose vertices are (0,0), (1,0) and (0,1). Let C be the curve obtained by proceeding around the boundary of A in an anti- clockwise direction. Prove İ}!").lx (ly İ)(2 dr dy. Pdr+Qdy That is, prove Green's Theorem for the triangle A. [Hint: the lecture notes have a proof for when A is a rectangle. So, the idea is is to give a similar proof where we have this...
7. Evaluate (6x - 6y+8) dx+(4 +9y +7) dy where C is the boundary of the triangle in the ry plane, wit h vertices at (0,0), (1,0)and (1,4) traversed once anticlockwise. (a) 10 (c) 20 (b)-8 (d) 8 10. Find the flux of F =-rit 2yj otward across the ellipse-+ -1. (a) 36π (b) 18m (c) o (d) 6π
7. Evaluate (6x - 6y+8) dx+(4 +9y +7) dy where C is the boundary of the triangle in the ry plane,...
(a) Let C be the line segment on the plane that starts from a point (xi,yi) to a different point (x2,Y2). Show that (b) Consider a simple polygon whose vertices are (2.1 , Й), (T2, Уг), . . . , (Xn, yn) if its boundary is traversed counterclockwise. Use Green's theorem to show that the area of this polygon is
(a) Let C be the line segment on the plane that starts from a point (xi,yi) to a different point...
3. (2 Points) Let Q be the quadrilateral in the ry-plane with vertices (1, 0), (4,0), (0, 1), (0,4). Consider 1 dA I+y Deda (a) Evaluate the integral using the normal ry-coordinates. (b) Consider the change of coordinates r = u-uv and y= uv. What is the image of Q under this change of coordinates?bi (c) Calculate the integral using the change of coordinates from the previous part. Change of Variables When working integrals, it is wise to choose a...
need 1-5
Midterm #3, Math 228 Each question is worth five points. 1. Let F(r.yzy). Let C be any curve that goes from A(-1,3,9) to B(1,6,-4). a) Show that F is conservative. b) Find a function φ such that ▽φ = F c) Use the result of b) to find Ic F Tds 2. Let F(z, y)-(2), and let C be the boundary of the square with vertices (1, 1). (-1,1). (-1,-1 traced out in the counter-clockwise direction. Find Jc...
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