Calc 3 Test 2 prep

14.3 Partial Derivatives

Definition of Partial Derivatives

A function of two variables, denoted as $f(x,y)$, exhibits the following partial derivatives, defined by limits:

• The partial derivative with respect to x:

  $f<em>x(x,y) = \frac{\text{lim}</em>{h \to 0} (f(x + h,y) - f(x,y))}{h}$

• The partial derivative with respect to y:

  $f<em>y(x,y) = \frac{\text{lim}</em>{h \to 0} (f(x,y + h) - f(x,y))}{h}$

Notation for First Order Partial Derivatives

• Common notations for the first order partial derivatives are:
  • $f_x(x,y)$
  • $\frac{\text{d} f}{\text{d} x}$
  • $\frac{\text{d} f}{\text{d} y}$

Rule of Thumb for Partial Differentiation

1. Treat $y$ as constant when differentiating with respect to $x$.
2. Treat $x$ as constant when differentiating with respect to $y$.

Examples of Partial Derivatives

Example 1:

Let $f(x,y) = 3x^2y^4 + 4xy^2$

• Calculating partial with respect to x:

  $f_x(x,y) = 6xy^4 + 4y^2$

• Calculating partial with respect to y:

  $f_y(x,y) = 12x^2y^3 + 8xy$
  Evaluation at Specific Points:

• At point (2,1):

  • $f_x(2,1) = 6(2)(1^4) + 4(1^2) = 12 + 4 = 16$
  • $f_y(2,1) = 12(2^2)(1^3) + 8(2)(1) = 48 + 16 = 64$
    Example 2:

  Let $f(x,y) = \text{sin}(x)$

• Calculating partial with respect to x,

  $f_x(x,y) = \text{cos}(x)$

• If $u = x$, then:

  $\frac{\text{d} f}{\text{d} u} = \text{cos}(x)$

Interpretation of Partial Derivatives

Given a surface of the form $z = f(x, y)$ where $f(a, b) = C$, the partial derivatives $f<em>x(a,b)$ and $f</em>y(a,b)$ represent the slopes of the tangent line to the curves $C<em>1$ and $C</em>2$ that intersect with the vertical planes defined by $y = b$ and $x = a$, respectively.
Curve C1
Intersects at $(a,b,C)$,

• Tangential Line Equation:

  $z - C = f_x(a,b)(x-a) \quad (\text{when } y=b)$
  Curve C2
  Intersects at $(a,b,C)$,

• Tangential Line Equation:

  $z - C = f_y(a,b)(y-b) \quad (\text{when } x=a)$

Interpretation of the Tangent Plane

The tangent plane at a point $(a,b,C)$ on the surface $z = f(x,y)$ has the equation:

$z - C = f<em>x(a,b)(x-a) + f</em>y(a,b)(y-b)$
This tangent plane approximates the function near $(a,b)$.

Functions of More Than Two Variables

For functions of more than two variables, e.g., $f(x,y,z)$, the partial derivatives follow the same rules as for two variables, treating the remaining variables as constants during differentiation.

Higher Order Partial Derivatives

For higher order partial derivatives, we can denote:

• The second-order partial derivative with respect to $x$:

  $f_{xx}(x,y) = \frac{\text{d}^2 f}{\text{d} x^2}$

• The mixed partial derivative:

  $f_{xy}(x,y) = \frac{\text{d}^2 f}{\text{d} x \text{d} y}$

Clairaut's Theorem

If $f$ is defined over a disc containing $(a,b)$ and both mixed derivatives $f<em>{xy}$ and $f</em>{yx}$ exist, then:

$f<em>{xy}(a,b) = f</em>{yx}(a,b)$

This theorem holds for most functions if they are continuously differentiable.

Implicit Differentiation

Consider a function defined implicitly as

$F(x,y,z) = 0$

To find the partial derivative $\frac{\text{d}z}{\text{d}x}$, we differentiate both sides with respect to $x$, treating $z$ as a function of $x$.

1. Apply Chain Rule:

   $F<em>x + F</em>y\frac{\text{dy}}{\text{dx}} + F_z\frac{\text{dz}}{\text{dx}} = 0$

2. Rearranging gives:

   $\frac{\text{dz}}{\text{dx}} = -\frac{F<em>x + F</em>y\frac{\text{dy}}{\text{dx}}}{F_z}$

Differentiation Rules for Functions of One Variable

1. Derivative of a constant is zero.
2. Derivative of $f(g(x))$ is $f'(g(x))g'(x)$ (Chain Rule).
3. Product rule: $(fg)' = f'g + fg'$.
4. Quotient rule: $\frac{g}{h}' = \frac{g'h - gh'}{h^2}$.
5. Higher-order derivatives are computed in a similar manner as basic derivatives.
6. Trigonometric derivatives:
   • $\frac{\text{d}}{\text{d}x}(\text{sin}(x)) = \text{cos}(x)$
   • $\frac{\text{d}}{\text{d}x}(\text{cos}(x)) = -\text{sin}(x)$

Tangent Planes & Linear Approximations

Linearization of a function

The linear approximation of a function near a point $(a,b)$ is given by:

• For a function $z = f(x,y)$, the approximation is:

  $z \approx f(a,b) + f<em>x(a,b)(x-a) + f</em>y(a,b)(y-b)$

This approximates $f(x,y)$ for $(x,y)$ near $(a,b)$.

Total Differential

For a function of two variables $z=f(x,y)$, the total differential $dz$ is defined as:

$dz = f<em>x(x,y)dx + f</em>y(x,y)dy$

• It represents the approximate change in $z$ for small changes $dx$ and $dy$ in $x$ and $y$.
• This is often used to approximate values or estimate errors.

The Chain Rule for Multivariable Functions

The Chain Rule is essential for differentiating composite functions where the input variables are themselves functions of other variables.

Case 1: $z = f(x,y)$ where $x = g(t)$ and $y = h(t)$

If $z = f(x,y)$ is a differentiable function of $x$ and $y$, and $x = g(t)$ and $y = h(t)$ are differentiable functions of $t$, then $z$ is a differentiable function of $t$, and:

$\frac{dz}{dt} = \frac{\partial f}{\partial x} \frac{dx}{dt} + \frac{\partial f}{\partial y} \frac{dy}{dt}$

Case 2: $z = f(x,y)$ where $x = g(s,t)$ and $y = h(s,t)$

If $z = f(x,y)$ is a differentiable function of $x$ and $y$, and $x = g(s,t)$ and $y = h(s,t)$ are differentiable functions of $s$ and $t$, then:

• Partial derivative with respect to $s$:

  $\frac{\partial z}{\partial s} = \frac{\partial f}{\partial x} \frac{\partial x}{\partial s} + \frac{\partial f}{\partial y} \frac{\partial y}{\partial s}$

• Partial derivative with respect to $t$:

  $\frac{\partial z}{\partial t} = \frac{\partial f}{\partial x} \frac{\partial x}{\partial t} + \frac{\partial f}{\partial y} \frac{\partial y}{\partial t}$

Case 3: General Chain Rule

If $u = f(x<em>1, x</em>2, \dots, x<em>n)$ and each $x</em>i = x<em>i(t</em>1, t<em>2, \dots, t</em>m)$, then the partial derivative of $u$ with respect to $t_j$ is:

$\frac{\partial u}{\partial t<em>j} = \frac{\partial u}{\partial x</em>1} \frac{\partial x<em>1}{\partial t</em>j} + \frac{\partial u}{\partial x<em>2} \frac{\partial x</em>2}{\partial t<em>j} + \dots + \frac{\partial u}{\partial x</em>n} \frac{\partial x<em>n}{\partial t</em>j}$

Directional Derivatives and the Gradient Vector

Directional Derivatives

The directional derivative of a function $f(x,y)$ in the direction of a unit vector $\mathbf{u} = \langle a, b \rangle$ is given by:

$D<em>\mathbf{u} f(x,y) = f</em>x(x,y)a + f_y(x,y)b$

• It represents the rate of change of $f$ in the direction of $\mathbf{u}$.

• For a function of three variables $f(x,y,z)$ and unit vector $\mathbf{u} = \langle a, b, c \rangle$:

  $D<em>\mathbf{u} f(x,y,z) = f</em>x(x,y,z)a + f<em>y(x,y,z)b + f</em>z(x,y,z)c$

The Gradient Vector

The gradient vector of a function $f(x,y)$ is denoted by $\nabla f(x,y)$ (read as "del $f$" or "gradient $f$") and is defined as:

$\nabla f(x,y) = \langle f<em>x(x,y), f</em>y(x,y) \rangle = \frac{\partial f}{\partial x} \mathbf{i} + \frac{\partial f}{\partial y} \mathbf{j}$

• The directional derivative can also be expressed using the dot product:

  $D_\mathbf{u} f(x,y) = \nabla f(x,y) \cdot \mathbf{u}$

Properties of the Gradient

• Maximum Rate of Increase: The maximum value of the directional derivative $D_\mathbf{u} f(x,y)$ is $|\nabla f(x,y)|$, and it occurs when $\mathbf{u}$ has the same direction as $\nabla f(x,y)$.
• Minimum Rate of Decrease: The minimum value is $-|\nabla f(x,y)|$, and it occurs when $\mathbf{u}$ has the opposite direction to $\nabla f(x,y)$.
• Level Curves/Surfaces: The gradient vector $\nabla f(x,y)$ is orthogonal (perpendicular) to the level curves $f(x,y) = k$ at a given point $(x,y)$. For functions of three variables, $\nabla f(x,y,z)$ is orthogonal to the level surfaces $f(x,y,z) = k$.

Maximum and Minimum Values

Critical Points

A critical point of a function $f(x,y)$ is a point $(a,b)$ in the domain of $f$ where either:

1. $f<em>x(a,b) = 0$ and $f</em>y(a,b) = 0$
2. Or, one or both of the partial derivatives do not exist.

The Second Derivatives Test

To classify critical points as local maxima, minima, or saddle points, we use the second derivatives test. Let $D$ be the discriminant defined as:

$D(x,y) = f<em>{xx}(x,y)f</em>{yy}(x,y) - [f_{xy}(x,y)]^2$

At a critical point $(a,b)$:

1. If $D(a,b) > 0$ and $f_{xx}(a,b) > 0$, then $f(a,b)$ is a local minimum.
2. If $D(a,b) > 0$ and $f_{xx}(a,b) < 0$, then $f(a,b)$ is a local maximum.
3. If $D(a,b) < 0$, then $f(a,b)$ is a saddle point (neither max nor min).
4. If $D(a,b) = 0$, the test is inconclusive.

Absolute Maximum and Minimum Values

To find the absolute maximum and minimum values of a continuous function $f$ on a closed, bounded set $D$:

1. Find the values of $f$ at the critical points of $f$ in $D$.
2. Find the extreme values of $f$ on the boundary of $D$. This often involves parameterizing the boundary or using methods like Lagrange Multipliers.
3. The largest of these values is the absolute maximum value, and the smallest is the absolute minimum value.

Polar Coordinates

Polar coordinates offer an alternative way to represent points in a two-dimensional plane using a distance from the origin and an angle from the positive x-axis. This system is particularly useful for problems with circular symmetry.

Definition and Conversion

• A point $P$ in the plane is represented by an ordered pair $(r, \theta)$, where:
  • $r$ is the distance from the origin (pole) to $P$.
  • $\theta$ is the angle (in radians) between the positive x-axis (polar axis) and the line segment from the origin to $P$.
• Conversion from Polar to Cartesian:
  • $x = r \cos \theta$
  • $y = r \sin \theta$
• Conversion from Cartesian to Polar:
  • $r^2 = x^2 + y^2$
  • $\tan \theta = \frac{y}{x} \quad (\text{if } x \neq 0)$

Area in Polar Coordinates

The area of a region bounded by polar curves $r = f(\theta)$ from $\theta = \alpha$ to $\theta = \beta$ is given by:

$A = \int_{\alpha}^{\beta} \frac{1}{2} r^2 d\theta$

Double Integrals in Polar Coordinates

To evaluate double integrals over regions with circular symmetry, converting to polar coordinates simplifies the integral. The differential area element $dA$ in Cartesian coordinates becomes $r dr d\theta$ in polar coordinates.

$\iint<em>R f(x,y) dA = \iint</em>D f(r \cos \theta, r \sin \theta) r dr d\theta$

Cylindrical Coordinates

Cylindrical coordinates extend polar coordinates into three dimensions by adding a $z$-coordinate. They are especially useful for problems with cylindrical symmetry.

Definition and Conversion

• A point $P$ in space is represented by $(r, \theta, z)$, where:
  • $(r, \theta)$ are the polar coordinates of the projection of $P$ onto the $xy$-plane.
  • $z$ is the Cartesian $z$-coordinate.
• Conversion from Cylindrical to Cartesian:
  • $x = r \cos \theta$
  • $y = r \sin \theta$
  • $z = z$
• Conversion from Cartesian to Cylindrical:
  • $r^2 = x^2 + y^2$
  • $\tan \theta = \frac{y}{x} \quad (\text{if } x \neq 0)$
  • $z = z$

Volume Element and Triple Integrals

In cylindrical coordinates, the differential volume element $dV$ in Cartesian coordinates becomes $r dz dr d\theta$ (or any permutation of $dz, dr, d\theta$). This Jacobian term $r$ is crucial for correct integration.

$\iiint<em>E f(x,y,z) dV = \iiint</em>G f(r \cos \theta, r \sin \theta, z) r dz dr d\theta$

Spherical Coordinates

Spherical coordinates provide another three-dimensional coordinate system, particularly suitable for problems with spherical symmetry.

Definition and Conversion

• A point $P$ in space is represented by $(\rho, \theta, \phi)$, where:
  • $\rho$ (rho) is the distance from the origin to $P$ ($\rho \ge 0$).
  • $\theta$ (theta) is the same angle as in cylindrical and polar coordinates ($0 \le \theta \le 2\pi$).
  • $\phi$ (phi) is the angle between the positive $z$-axis and the line segment from the origin to $P$ ($0 \le \phi \le \pi$).
• Conversion from Spherical to Cartesian:
  • $x = \rho \sin \phi \cos \theta$
  • $y = \rho \sin \phi \sin \theta$
  • $z = \rho \cos \phi$
• Conversion from Cartesian to Spherical:
  • $\rho^2 = x^2 + y^2 + z^2$
  • $\tan \theta = \frac{y}{x} \quad (\text{if } x \neq 0)$
  • $\cos \phi = \frac{z}{\rho} = \frac{z}{\sqrt{x^2+y^2+z^2}}$

Volume Element and Triple Integrals

In spherical coordinates, the differential volume element $dV$ in Cartesian coordinates becomes $\rho^2 \sin \phi d\rho d\phi d\theta$. The Jacobian is $\rho^2 \sin \phi$.

$\iiint<em>E f(x,y,z) dV = \iiint</em>H f(\rho \sin \phi \cos \theta, \rho \sin \phi \sin \theta, \rho \cos \phi) \rho^2 \sin \phi d\rho d\phi d\theta$

Triple Integration

Triple integrals are used to integrate functions of three variables over a three-dimensional region. They are fundamental for calculating volumes, masses, centroids, and moments of inertia for solids.

Definition

The triple integral of a function $f(x,y,z)$ over a solid region $E$ is defined as:

$\iiint<em>E f(x,y,z) dV = \text{lim}</em>{l,m,n \to \infty} \sum<em>{i=1}^l \sum</em>{j=1}^m \sum<em>{k=1}^n f(x</em>{ijk}^<em>, y<em>{ijk}^</em>, z</em>{ijk}^*) \Delta V$

where $\Delta V$ is the volume of a small rectangular box and $(x<em>{ijk}^<em>, y</em>{ijk}^</em>, z_{ijk}^*)$ is a sample point within that box.

Iterated Integrals for Triple Integrals

Similar to double integrals, triple integrals are typically evaluated as iterated integrals. For a function $f$ over a region $E$ described by:

$E = { (x,y,z) \mid a \le x \le b, g<em>1(x) \le y \le g</em>2(x), h<em>1(x,y) \le z \le h</em>2(x,y) }$

The triple integral is:

$\iiint<em>E f(x,y,z) dV = \int</em>a^b \int<em>{g</em>1(x)}^{g<em>2(x)} \int</em>{h<em>1(x,y)}^{h</em>2(x,y)} f(x,y,z) dz dy dx$

• The order of integration can be changed, and there are six possible orders ($dz dy dx, dz dx dy, dy dz dx, dy dx dz, dx dy dz, dx dz dy$), depending on the region $E$.

Applications

• Volume: If $f(x,y,z) = 1$, the triple integral gives the volume of the region $E$:

  $V = \iiint_E 1 dV$

• Mass: If $\rho(x,y,z)$ is the density function of a solid object $E$, its total mass $M$ is:

  $M = \iiint_E \rho(x,y,z) dV$

• Centroid: The coordinates of the centroid $(\bar{x}, \bar{y}, \bar{z})$ are given by:

  • $\bar{x} = \frac{1}{M} \iiint_E x \rho(x,y,z) dV$
  • $\bar{y} = \frac{1}{M} \iiint_E y \rho(x,y,z) dV$
  • $\bar{z} = \frac{1}{M} \iiint_E z \rho(x,y,z) dV$

Change of Variables in Triple Integrals

When integrating over complex regions in Cartesian coordinates, changing to cylindrical or spherical coordinates can simplify computations. The conversion involves a Jacobian determinant, which accounts for the scaling of the volume elements:

• Cylindrical: $dV = r dr d\theta dz$
• Spherical: $dV = \rho^2 \sin \phi d\rho d\phi d\theta$