Partial Differentiation

• What is the definition of a partial derivative?

  It measures the rate of change of a function.
• How is the slope defined for a function of a single variable?

  By the limit of the difference quotient.
• What does the z-axis represent in a multivariate function f(x,y)?

  The function value z = f(x,y).
• What notation is used to denote the partial derivative with respect to x?

  ∂f/∂x.
• What happens to the other variable when calculating a partial derivative?

  The other variable is treated as a constant.
• What is the formula for the partial derivative with respect to x?

  ∂f/∂x = lim(∆x→0) [f(x + ∆x,y) - f(x,y)]/∆x.
• What is the formula for the partial derivative with respect to y?

  ∂f/∂y = lim(∆y→0) [f(x,y + ∆y) - f(x,y)]/∆y.
• What do higher-order partial derivatives represent?

  They represent the rate of change of partial derivatives.
• What is the notation for the second-order partial derivative with respect to x?

  ∂²f/∂x².
• What is the notation for mixed partial derivatives?

  ∂²f/∂x∂y.
• What does Schwarz’s Theorem state about mixed partial derivatives?

  They are symmetric if continuous.
• How do you calculate the gradient vector for a function f(x,y)?

  By combining partial derivatives as a vector.
• What does the gradient vector indicate about a function?

  It shows the direction of steepest ascent.
• What is the formula for the gradient vector?

  ∇f(x,y) = ∂f/∂x i + ∂f/∂y j.
• In what space does the gradient vector exist?

  In the 2D space of the xy-plane.
• How does the gradient vector change with position (x,y)?

  It depends on the values of x and y.
• What is the total differential for a function of two variables?

  df = (∂f/∂x)dx + (∂f/∂y)dy.
• What does the total differential represent?

  The change in the function value for small changes.
• How do you express the total derivative of a function f(x,y) with respect to t?

  df/dt = (∂f/∂x)(dx/dt) + (∂f/∂y)(dy/dt).
• What is the significance of the direction of dr in the total derivative?

  It determines the rate of change of f.
• What happens if the direction of dr is perpendicular to the gradient vector?

  The slope of the function is zero.
• What are the steps to find the total differential of a function f(x,y)?

  1. Calculate partial derivatives ∂f/∂x and ∂f/∂y.
  2. Use the formula df = (∂f/∂x)dx + (∂f/∂y)dy.
  3. Interpret df as the change in f for small dx and dy.
• What are the geometric interpretations of the total derivative in terms of the gradient vector?

  • Slope in the direction of the gradient is maximal.
  • Slope in the opposite direction is negatively maximal.
  • Slope is zero when perpendicular to the gradient.
• What are the partial derivatives of T with respect to t, x, and y?

  ∂T/∂t = 1/t, ∂T/∂x = e^{-y}, ∂T/∂y = -xe^{-y}
• What are the derivatives of x(t) and y(t) with respect to t?

  dx/dt = a, dy/dt = 3bt^2
• How is the total derivative dT/dt expressed?

  dT/dt = 1/t + e^{-y}a - xe^{-y}3bt^2
• What geometric interpretations can be inferred from the total derivative?

  • Slope in direction of \( \hat{e} \)
  • Maximal slope when \( \hat{e} \) aligns with gradient
  • Negatively maximal slope when \( \hat{e} \) opposes gradient
  • Zero slope when \( \hat{e} \) is perpendicular to gradient
  • Zero slope at stationary points
• What is the condition for a differential df to be exact?

  ∂A/∂y = ∂B/∂x
• What is an inexact differential denoted as?

  ¯df
• What do exact differentials correspond to?

  Changes of state functions independent of path
• What is the formula for the exactness condition?

  df = A(x,y)dx + B(x,y)dy
• How can we determine if a differential is exact?

  Check if second-order derivatives are symmetric
• What is the significance of the second-order derivatives being symmetric?

  It indicates the function is well-behaved
• What are the conditions for local maxima, minima, and saddle points?

  • Local minimum: \( f_{xx} > 0, f_{yy} > 0, f_{xx}f_{yy} > f_{xy}^2 \)
  • Local maximum: \( f_{xx} < 0, f_{yy} < 0, f_{xx}f_{yy} > f_{xy}^2 \)
  • Saddle point: \( f_{xx}f_{yy} < f_{xy}^2 \)
  • Undetermined: \( f_{xx}f_{yy} = f_{xy}^2 \)
• What is the formula for the total differential df?

  df = A(x,y)dx + B(x,y)dy
• How do you find stationary points of a function f(x,y)?

  Set partial derivatives to zero
• What does the total differential vanish at a stationary point indicate?

  Function value remains constant in any direction
• What is the first step to determine the nature of a stationary point?

  Calculate the second-order derivatives
• What is the nature of the stationary point if \( f_{xx} > 0 \) and \( f_{yy} > 0 \)?

  It is a local minimum
• What does it mean if \( f_{xx}f_{yy} < f_{xy}^2 \)?

  It indicates a saddle point