Vector Operations and Their Applications

The dot product of two vectors is a scalar, found via v₁u₁ + v₂u₂ + v₃u₃

Adding two vectors results in a new vector.

Two vectors are orthogonal if their dot product is zero.

x = x₁ + at, y = y₁ + bt, z = z₁ + ct; (x₁, y₁, z₁) is a point on line and is the direction vector

Ax + By + Cz = D. A, B, C are coefficients of directional vector normal to plane

(x-h)² + (y-k)² + (z-l)² = r². (h, k, l) is center, r is radius

x² / a² + y² / b² - z² / c² = 1; three axes, one negative, and two positive

The cross product of two vectors is a vector; | i j k | | u1 u2 u3 | |v1 v2 v3|

A function that maps a real number (t) to a vector in space. It is represented as (\vec{r}(t) = \langle f(t), g(t), h(t)\rangle) or (\vec{r}(t) = \vec{f}(t) + \vec{g}(t) + \vec{h}(t)), where f(t), g(t), and h(t) are scalar functions.

The derivative of a vector-valued function (\vec{r}(t)) at a point (t_0), denoted by (\vec{r}'(t_0)), gives the direction of the tangent line to the curve at that point.

The length of a curve defined by a vector-valued function (\vec{r}(t)) from (t = a) to (t = b) is calculated by integrating the magnitude of the derivative of the function over the interval: (L = \int_a^b ||\vec{r}'(t)|| , dt)

A measure of how sharply a curve bends at a given point. It is calculated by the formula (K = \frac{||\vec{T}'(t)||}{||\vec{r}'(t)||} = \frac{||\vec{r}'(t) \times \vec{r}''(t)||}{||\vec{r}'(t)||^3}), where (\vec{T}(t)) is the unit tangent vector.

For a function of two variables, z = f(x, y), a level curve is a set of points (x, y) where the function has a constant value. It's like a contour line on a map showing points with the same elevation.

This test determines if a limit exists by checking if the function approaches the same value along different paths towards the point. If the limits differ for different paths, the overall limit does not exist.

A function is continuous at a point if its limit at that point exists and equals the value of the function at that point. This means there are no jumps or breaks in the function.

The rate of change of a multivariable function with respect to one variable while holding others constant. For example, fx finds how 'f' changes as 'x' changes while 'y' is kept fixed.

Used to find the derivative of a function where the variables depend on other variables. It combines the derivatives of individual functions to obtain the derivative of the whole function.

The directional derivative measures the rate of change of a function in a specific direction. It tells you how fast the function changes along that particular direction.

A method used to find maximum and minimum values of a function f(x, y, z) subject to a constraint g(x, y, z) = 0, by finding solutions to the equation ∇f = λ∇g, along with g(x, y, z) = 0.

A method to calculate the volume under a surface z = f(x, y) over a given region in the xy-plane.

The average value of a function f(x, y) over a region R in the xy-plane is calculated by dividing the double integral of f(x, y) over R by the area of R.

A method to calculate double integrals over regions that are easier to describe in polar coordinates, using the transformations x = r cos(θ) and y = r sin(θ), and dA = r dr dθ.

A method to calculate the volume of a solid or integrate functions over 3D regions.

A coordinate system used for triple integrals, where x = r cos(θ), y = r sin(θ), and z = z, with dV = r dr dθ dz. Useful for regions with rotational symmetry around the z-axis.

A coordinate system used for triple integrals, where x = ρ sin(φ) cos(θ), y = ρ sin(φ) sin(θ), z = ρ cos(φ), and dV = ρ² sin(φ) dρ dφ dθ. Useful for spheres and hemispherical regions.

When converting from rectangular coordinates to polar coordinates in a double integral, you use the relationships x = r cos(θ) and y = r sin(θ) to change the integrand and the limits of integration to reflect the polar representation of the region.

The total mass of a solid object with varying density is calculated by integrating the density function over the solid's volume.

A vector field assigns a vector (magnitude and direction) to each point in space.

Calculates the integral of a scalar function along a curve in space, considering the curve's length.

Integrates a vector field along a curve, considering the dot product with the curve's tangent vector.

A vector field where the line integral between any two points is independent of the path taken.

A measure of how much a vector field is 'outflowing' or 'expanding' at a point. It's like measuring the 'source' of a fluid.

A measure of how much a vector field is 'rotating' or 'twisting' at a point. It indicates the 'circulation' of the field.

Calculates the total amount of a vector field flowing through a given surface. It's like measuring the 'flux' of a fluid through a membrane.

Relates the surface integral of the curl of a vector field to the line integral of the field around the boundary of the surface.

Relates the flux of a vector field through a closed surface to the triple integral of the divergence of the field over the enclosed volume.