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Given z =f(x, y) and w = g(x, y) such that a/ax = aw/ay and az/ay-みv/ar. If θι and θ2 are two mutually perpendicular directions, show that at any point FOx, y), as/as, = aw/as, and as/as, =-aw/a...
ax az . Letſ be a differentiable function of one variable, and let w = f(p),...
ax az . Letſ be a differentiable function of one variable, and let w = f(p), where p = (x2 + y2 + 2)/2. Show that dw ay · Let z = f(x - y. y - x). Show that az/ax + az/ay=0. Let f be a differentiable function of three variables and sup- pose that w = Sex - y. y - 2.2 - x). Show that aw ду az Page 1 / 1 aw aw ax + +...
Please show all work Show that F= + X0, +99, + (x+y). (where Problem 1. V...
Please show all work
Show that F= + X0, +99, + (x+y). (where Problem 1. V -0) satisfies the equation v'(V°F) - 0 videnotes the operator ax ay? Problem 2. Given: Eu - (1-0) ReZ-(1 +vby Im Z Ev - 2 Im Z - (1 +) y Re Z Show that these expressions for the displacements, u and y, are consistent with Hooke's law equations. (Plane stress): Ex -0,-voy E, -, -vo E_Y = ? 2(1+0) Problem 3. For a...
Im having problems understanding whats happening and why. 27. An xz-cylinder in 3-space is a surface given by an equation f(x,z) -0 in x and z alone; its section by any plane y -c perpendicular to the...
Im having problems
understanding whats happening and why.
27. An xz-cylinder in 3-space is a surface given by an equation f(x,z) -0 in x and z alone; its section by any plane y -c perpendicular to the y-axis is always the same xz- curve. Show that if F - z2i + xy2j + xz k, then ^ F dr for any simple closed curve C lying on an xz-cylinder. (Use Stokes' theorem)
27. An xz-cylinder in 3-space is a surface...
5) Let P(1,2,2) be a point, and f(x,y,z) and g(x,y,z) be two differentiable functions satisfying the...
5) Let P(1,2,2) be a point, and f(x,y,z) and g(x,y,z) be two differentiable functions satisfying the following conditions. 1) f(P)=1 and g(P)=4 og IT) = -2 Oz IP III) The direction in which f increases most rapidly at the point Pis ū=4i - +8k , and the derivative in this direction is 3. IV) Equation of the plane tangent to the surface f(x,y,z)+3g(x,y,z)=13 at the int P is x+4y + 5z =19 According to this, calculate og Ox . (20P)
1. (18 pts.) Short answer: a) Given that f (u,v,w,x, y,z) is a real valued function,...
1. (18 pts.) Short answer: a) Given that f (u,v,w,x, y,z) is a real valued function, what dimension is its graph in? What dimension are its level curves in? (2,1,3). b) Given: g(x,y,z)= x? In (xy+z) Find the direction of maximum ascent c) Find I using integration or geometry: I = 6 dx dy. S: 146 d) Describe and/or draw the region: R= P ={(2,0,0)|9=%;05057; p=17}
If a function has more than one independent variables, e.g. y = f(x,z), then the change...
If a function has more than one independent variables, e.g. y = f(x,z), then the change in y is generalized to Ay= 4x + Az where the partial derivative oy is the derivative of y with respect tox treating all other independent variables (in this case z) as constant. Example: The area of a rectangle is A = LW. The two independent variables are the length Land width W of a rectangle. Then the average area is Aav = Lay...
(1 point) Horizontal cross-sections of the vector fields F(x, y, z) and G(x, y, z) are...
(1 point) Horizontal cross-sections of the vector fields F(x, y, z) and G(x, y, z) are given in the figure. Each vector field has zero z-component (i.e., all of its vectors are horizontal) and is independent of 2 (i.e., is the same in every horizontal plane). You may assume that the graphs of these vector fields use the same scale. (a) Are div(F) and div(G) positive, negative, or zero at the origin? Be sure you can explain your answer. At...
Problem 5. Find the local marimum and minimum values and saddle point(s) of the functions: i)...
Problem 5. Find the local marimum and minimum values and saddle point(s) of the functions: i) f(x,y) = x2 + xy + y2 + y. a) f(x, y) = (x - y)(1 - x). ui) (Optional) f(0,y) = xy +e-zy. Note that the critical points are (2,0) and (0,y) and that f(x,0) = f(0, y) = 1. However, from Math 110, we can show that the function gw) = w+e-w has an absolute mim at w = 0i.e., g(w) >...
Problem 3 (LrTrmations). (a) Give an example of a fuction R R such that: f(Ax)-Af(x), for...
Problem 3 (LrTrmations). (a) Give an example of a fuction R R such that: f(Ax)-Af(x), for all x € R2,AG R, but is not a linear transformation. (b) Show that a linear transformation VWfrom a one dimensional vector space V is com- pletely determined by a scalar A (e) Let V-UUbe a direet sum of the vector subspaces U and Ug and, U2 be two linear transformations. Show that V → W defined by f(m + u2)-f1(ul) + f2(u2) is...
(1 point In general for a non-homogeneous problem y' + p(x) +(z) = f() assume that...
(1 point In general for a non-homogeneous problem y' + p(x) +(z) = f() assume that y. is a fundamental set of solutions for the homogeneous problemy" p(x) + (2) 0. Then the formula for the particular solution using the method of variation of parameters is where (z)/ and ()/() where W() is the Wronskian given by the determinant W (2) (2) W2) 31(2)/(2) dr. NOTE When evaluating these indefinite integrals we take the W(2) So we have the de...
ax az . Letſ be a differentiable function of one variable, and let w = f(p), where p = (x2 + y2 + 2)/2. Show that dw ay · Let z = f(x - y. y - x). Show that az/ax + az/ay=0. Let f be a differentiable function of three variables and sup- pose that w = Sex - y. y - 2.2 - x). Show that aw ду az Page 1 / 1 aw aw ax + +...
Please show all work
Show that F= + X0, +99, + (x+y). (where Problem 1. V -0) satisfies the equation v'(V°F) - 0 videnotes the operator ax ay? Problem 2. Given: Eu - (1-0) ReZ-(1 +vby Im Z Ev - 2 Im Z - (1 +) y Re Z Show that these expressions for the displacements, u and y, are consistent with Hooke's law equations. (Plane stress): Ex -0,-voy E, -, -vo E_Y = ? 2(1+0) Problem 3. For a...
Im having problems
understanding whats happening and why.
27. An xz-cylinder in 3-space is a surface given by an equation f(x,z) -0 in x and z alone; its section by any plane y -c perpendicular to the y-axis is always the same xz- curve. Show that if F - z2i + xy2j + xz k, then ^ F dr for any simple closed curve C lying on an xz-cylinder. (Use Stokes' theorem)
27. An xz-cylinder in 3-space is a surface...
5) Let P(1,2,2) be a point, and f(x,y,z) and g(x,y,z) be two differentiable functions satisfying the following conditions. 1) f(P)=1 and g(P)=4 og IT) = -2 Oz IP III) The direction in which f increases most rapidly at the point Pis ū=4i - +8k , and the derivative in this direction is 3. IV) Equation of the plane tangent to the surface f(x,y,z)+3g(x,y,z)=13 at the int P is x+4y + 5z =19 According to this, calculate og Ox . (20P)
1. (18 pts.) Short answer: a) Given that f (u,v,w,x, y,z) is a real valued function, what dimension is its graph in? What dimension are its level curves in? (2,1,3). b) Given: g(x,y,z)= x? In (xy+z) Find the direction of maximum ascent c) Find I using integration or geometry: I = 6 dx dy. S: 146 d) Describe and/or draw the region: R= P ={(2,0,0)|9=%;05057; p=17}
If a function has more than one independent variables, e.g. y = f(x,z), then the change in y is generalized to Ay= 4x + Az where the partial derivative oy is the derivative of y with respect tox treating all other independent variables (in this case z) as constant. Example: The area of a rectangle is A = LW. The two independent variables are the length Land width W of a rectangle. Then the average area is Aav = Lay...
(1 point) Horizontal cross-sections of the vector fields F(x, y, z) and G(x, y, z) are given in the figure. Each vector field has zero z-component (i.e., all of its vectors are horizontal) and is independent of 2 (i.e., is the same in every horizontal plane). You may assume that the graphs of these vector fields use the same scale. (a) Are div(F) and div(G) positive, negative, or zero at the origin? Be sure you can explain your answer. At...
Problem 5. Find the local marimum and minimum values and saddle point(s) of the functions: i) f(x,y) = x2 + xy + y2 + y. a) f(x, y) = (x - y)(1 - x). ui) (Optional) f(0,y) = xy +e-zy. Note that the critical points are (2,0) and (0,y) and that f(x,0) = f(0, y) = 1. However, from Math 110, we can show that the function gw) = w+e-w has an absolute mim at w = 0i.e., g(w) >...
Problem 3 (LrTrmations). (a) Give an example of a fuction R R such that: f(Ax)-Af(x), for all x € R2,AG R, but is not a linear transformation. (b) Show that a linear transformation VWfrom a one dimensional vector space V is com- pletely determined by a scalar A (e) Let V-UUbe a direet sum of the vector subspaces U and Ug and, U2 be two linear transformations. Show that V → W defined by f(m + u2)-f1(ul) + f2(u2) is...
(1 point In general for a non-homogeneous problem y' + p(x) +(z) = f() assume that y. is a fundamental set of solutions for the homogeneous problemy" p(x) + (2) 0. Then the formula for the particular solution using the method of variation of parameters is where (z)/ and ()/() where W() is the Wronskian given by the determinant W (2) (2) W2) 31(2)/(2) dr. NOTE When evaluating these indefinite integrals we take the W(2) So we have the de...
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