Partial Derivatives in Multivariable Calculus
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Questions and Answers

The derivative of u with respect to x, treating y as constant is called the ______ of u w.r.t x.

partial derivative

What does the slope of the curve z = f(x,y) at the point P(x0,y0, f(x0,y0)) in the plane y = y0 represent?

The value of the partial derivative of f with respect to x at (x0, y0).

The partial derivative of a function of multiple independent variables is computed by treating all remaining independent variables as constant.

True

What is the total differential of a function u = f(x,y) with respect to t, where x = x(t) and y = y(t)?

<p>du/dt = (du/dx)(dx/dt) + (du/dy)(dy/dt)</p> Signup and view all the answers

What is the Jacobian of two functions u and v with respect to x and y?

<p>The Jacobian of u and v with respect to x and y is denoted by J = (u,v)/d(x,y) = (ux uy)/(vx vy).</p> Signup and view all the answers

What is the condition for two functions u and v to be considered functionally dependent?

<p>Their Jacobian is equal to 0.</p> Signup and view all the answers

What is the formula for the Taylor's series of a function f(x,y) expanded around the point (a,b)?

<p>f(x,y) = f(a,b) + (x-a)df(a,b)/dx + (y-b)df(a,b)/dy + 1/2![(x-a)²d²f(a,b)/dx² + 2(x-a)(y-b)d²f(a,b)/dxdy + (y-b)²d²f(a,b)/dy²] + ...</p> Signup and view all the answers

What is the formula for the Maclaurin's series of a function f(x,y) expanded around the point (0,0)?

<p>f(x,y) = f(0,0) + xdf(0,0)/dx + ydf(0,0)/dy + 1/2![(x)²d²f(0,0)/dx² + 2xy d²f(0,0)/dxdy + (y)²d²f(0,0)/dy²] + ...</p> Signup and view all the answers

What are the necessary conditions for a function f(x,y) to have a maximum or minimum at the point (a,b)?

<p>fx(a,b) = 0 and fy(a,b) = 0.</p> Signup and view all the answers

What condition defines a saddle point for a function z = f(x,y) at the point (x0, y0, f(x0, y0))?

<p>fx(x0, y0) = 0, fy(x0, y0) = 0, but f(x0, y0) does not have a local extremum.</p> Signup and view all the answers

Study Notes

Partial Derivatives

  • Let u = f(x, y) be a function of two variables x and y. If y is constant and x varies, u is a function of x only.
  • The derivative of u with respect to x, treating y as constant, is the partial derivative of u with respect to x, denoted as ∂u/∂x or ux.
  • Similarily, the partial derivative of u with respect to y (treating x as constant), is denoted as ∂u/∂y or uy.
  • These partial derivatives also depend on both x and y so can further be differentiated partially.

Geometric Interpretation

  • The slope of the curve z = f(x, y) at a point P(x₀, y₀, f(x₀, y₀)) in the plane y = y₀ is the partial derivative of f with respect to x at (x₀, y₀).

Partial Derivatives of Composite Functions

  • If u = f(x, y) where x = g(s, t) and y = h(s, t), then ∂u/∂s and ∂u/∂t can be calculated using the chain rule.

Jacobians

  • If u and v are functions of two independent variables x and y, the Jacobian of u and v with respect to x and y is denoted as ∂(u, v)/∂(x, y).
  • It's a determinant: ∂(u, v)/∂(x, y) = | (∂u/∂x) (∂v/∂y) - (∂u/∂y) (∂v/∂x) |
  • The Jacobian has properties useful in various applications (function dependency, transformations).

Taylor's Series for Functions of Two Variables

  • The Taylor series expansion for a function f(x, y) about a point (a, b) includes terms involving partial derivatives.

Maclaurin's Series for Functions of Two Variables

  • The Maclaurin series expansion for a function f(x, y) about the point (0, 0) involves partial derivatives.

Maxima and Minima of Functions of Two Variables

  • A function z = f(x, y) has a maximum or minimum at (a, b) if f(a + h, b + k) < or > f(a, b) for all small h and k.
  • Necessary conditions for a maximum or minimum are fx(a, b) = 0 and fy(a, b) = 0.
  • Second-order partial derivatives (r, s, t) help determine if a point is a maximum, minimum, or saddle point.

Errors and Approximations

  • Error analysis in calculations involving functions of several variables considers the absolute, relative, and percentage errors for input variables leading to errors in computed results.

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Unit 2 Multivariable Calculus PDF

Description

This quiz explores the concepts of partial derivatives, their geometric interpretation, and their applications in composite functions. You'll learn about the notation used for partial derivatives and the role of Jacobians in multivariable functions. Test your understanding of these essential calculus topics!

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