12.3 Surface area of a function graph

Calculating surface area is a key application of double integrals. We'll learn how to find the area of a function's graph over a region using partial derivatives and the surface area formula.

This topic builds on our knowledge of double integrals and partial derivatives. We'll see how these concepts come together to measure the size of curved surfaces in three-dimensional space.

Surface Area Formula and Double Integral

Calculating Surface Area Using the Surface Area Formula

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• The surface area of a function graph $z=f(x,y)$ over a region $R$ in the $xy$-plane can be calculated using the surface area formula:
  • $\iint_R \sqrt{1+(\frac{\partial f}{\partial x})^2 + (\frac{\partial f}{\partial y})^2} \, dA$
  • This formula involves a double integral over the region $R$
  • The integrand contains the partial derivatives of $f$ with respect to $x$ and $y$
• The double integral for surface area represents the sum of the areas of infinitesimal surface elements over the region $R$
  • Each surface element is an approximation of a small portion of the surface
  • The surface element is denoted by $dS$ and is related to the area element $dA$ in the $xy$-plane

Jacobian and Its Role in Surface Area Calculation

• The Jacobian, denoted as $\sqrt{1+(\frac{\partial f}{\partial x})^2 + (\frac{\partial f}{\partial y})^2}$, appears in the surface area formula
  • It accounts for the stretching or compression of the surface element when projected onto the $xy$-plane
  • The Jacobian measures the ratio of the surface element area to the corresponding area element in the $xy$-plane
• The presence of the Jacobian in the surface area formula ensures that the double integral accurately represents the surface area
  • It adjusts the contribution of each surface element based on the local geometry of the surface
  • Without the Jacobian, the double integral would not provide the correct surface area measurement

Parametric Surfaces

Representing Surfaces Using Parametric Equations

• A parametric surface is a surface described by parametric equations in terms of two parameters, typically denoted as $u$ and $v$
  • The equations are of the form: $x=x(u,v)$, $y=y(u,v)$, $z=z(u,v)$
  • The parameters $u$ and $v$ vary over a certain domain, usually a rectangular region in the $uv$-plane
• Parametric equations provide a convenient way to represent complex surfaces that may be difficult to express as a single function $z=f(x,y)$
  • Examples of parametric surfaces include spheres, cylinders, and tori (donut-shaped surfaces)
  • Parametric representations allow for more flexibility in describing the shape and geometry of surfaces

Partial Derivatives of Parametric Surfaces

• To calculate various properties of parametric surfaces, such as tangent planes and surface area, partial derivatives are used
  • Partial derivatives of the parametric equations with respect to $u$ and $v$ are denoted as $\frac{\partial x}{\partial u}$, $\frac{\partial y}{\partial u}$, $\frac{\partial z}{\partial u}$, $\frac{\partial x}{\partial v}$, $\frac{\partial y}{\partial v}$, and $\frac{\partial z}{\partial v}$
  • These partial derivatives represent the rates of change of the coordinates $(x,y,z)$ with respect to the parameters $u$ and $v$
• The partial derivatives of parametric surfaces are used to determine important geometric properties
  • They are involved in the calculation of tangent vectors, normal vectors, and the surface area of parametric surfaces
  • Understanding how to compute and interpret partial derivatives is crucial for analyzing parametric surfaces

Normal Vectors

Definition and Significance of Normal Vectors

• A normal vector is a vector that is perpendicular to a surface at a given point
  • It is denoted as $\vec{n}$ and is typically represented as a unit vector (a vector with length 1)
  • The direction of the normal vector indicates the orientation of the surface at that point
• Normal vectors have various applications in mathematics and physics
  • They are used to determine the direction of force acting on a surface (such as in fluid dynamics)
  • Normal vectors are also employed in computer graphics to calculate lighting and shading effects on surfaces

Calculating Normal Vectors for Parametric Surfaces

• For parametric surfaces, the normal vector at a point can be calculated using the cross product of the partial derivatives
  • The partial derivatives $\frac{\partial \vec{r}}{\partial u}$ and $\frac{\partial \vec{r}}{\partial v}$ (where $\vec{r}(u,v) = (x(u,v), y(u,v), z(u,v))$) represent tangent vectors to the surface
  • The cross product of these partial derivatives gives a vector perpendicular to both tangent vectors, which is the normal vector
  • The formula for the normal vector is: $\vec{n} = \frac{\partial \vec{r}}{\partial u} \times \frac{\partial \vec{r}}{\partial v}$
• Normalizing the resulting vector (dividing it by its magnitude) yields a unit normal vector
  • Unit normal vectors are often preferred for consistency and simplicity in calculations
  • The direction of the unit normal vector can be chosen to point outward or inward, depending on the convention used