What is the domain of a multivariable function?
The set of all input values where the function is defined.
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| Term | Definition |
|---|---|
What is the domain of a multivariable function? | The set of all input values where the function is defined. |
What must you check when finding domain? | Denominators cannot be 0, even roots need nonnegative inputs, and logarithms need positive inputs. |
What is a level curve? | A curve defined by f(x,y)=c for some constant c. |
What is a level surface? | A surface defined by f(x,y,z)=c for some constant c. |
What is a partial derivative with respect to x? | The derivative found by treating all other variables as constants: fx = ∂f/∂x. |
What is a partial derivative with respect to y? | The derivative found by treating all other variables as constants: fy = ∂f/∂y. |
What are second partial derivatives? | Derivatives such as fxx, fyy, fxy, and fyx. |
When are mixed partials equal? | If the mixed partials are continuous, then fxy = fyx. |
Chain rule for z=f(x,y), x=x(t), y=y(t) | dz/dt = fx(dx/dt) + fy(dy/dt) |
Chain rule for z=f(x,y), x=x(s,t), y=y(s,t): ∂z/∂s | ∂z/∂s = fx(∂x/∂s) + fy(∂y/∂s) |
Chain rule for z=f(x,y), x=x(s,t), y=y(s,t): ∂z/∂t | ∂z/∂t = fx(∂x/∂t) + fy(∂y/∂t) |
What is the gradient of f(x,y)? | ∇f = <fx, fy> |
What is the gradient of f(x,y,z)? | ∇f = <fx, fy, fz> |
What does the gradient vector point toward? | The direction of greatest increase. |
What is the maximum rate of change of a function? | |∇f| |
What is the directional derivative formula? | Du f = ∇f · u, where u is a unit vector. |
Important condition for directional derivative | The direction vector must be a unit vector. |
How do you turn a vector v into a unit vector? | u = v / |v| |
Magnitude of vector <a,b> | |v| = sqrt(a^2 + b^2) |
Magnitude of vector <a,b,c> | |v| = sqrt(a^2 + b^2 + c^2) |
Relationship between gradient and level curves | The gradient is perpendicular to level curves. |
Relationship between gradient and level surfaces | The gradient is perpendicular to level surfaces. |
Tangent plane to z=f(x,y) at (x0,y0) | z - z0 = fx(x0,y0)(x - x0) + fy(x0,y0)(y - y0), where z0=f(x0,y0) |
What is the linearization formula? | L(x,y) = f(x0,y0) + fx(x0,y0)(x - x0) + fy(x0,y0)(y - y0) |
What is the normal vector to the surface F(x,y,z)=c? | ∇F = <Fx, Fy, Fz> |
Tangent plane to F(x,y,z)=c at (x0,y0,z0) | Fx(x0,y0,z0)(x-x0) + Fy(x0,y0,z0)(y-y0) + Fz(x0,y0,z0)(z-z0) = 0 |
How do you find critical points of f(x,y)? | Solve fx=0 and fy=0, and also check where derivatives do not exist. |
Second derivative test discriminant | D = fxx fyy - (fxy)^2 |
Second derivative test: local minimum | If D > 0 and fxx > 0, then local minimum. |
Second derivative test: local maximum | If D > 0 and fxx < 0, then local maximum. |
Second derivative test: saddle point | If D < 0, then saddle point. |
Second derivative test: inconclusive | If D = 0, the test is inconclusive. |
How do you find absolute max/min on a closed bounded region? | Check interior critical points, check the boundary, then compare all function values. |
What theorem guarantees absolute max/min on a closed bounded region? | Extreme Value Theorem. |
What is the Lagrange multipliers equation? | ∇f = λ∇g |
What else must be used with Lagrange multipliers? | The constraint equation g(x,y,z)=c |
Lagrange multipliers in two variables | fx = λgx, fy = λgy, and g(x,y)=c |
What is a double integral? | An integral of the form ∬R f(x,y) dA over a region R. |
Area of a region using a double integral | A = ∬R 1 dA |
Volume under z=f(x,y) where f≥0 | V = ∬R f(x,y) dA |
Double integral over a Type I region | ∬R f(x,y) dA = ∫[a to b] ∫[g1(x) to g2(x)] f(x,y) dy dx |
Double integral over a Type II region | ∬R f(x,y) dA = ∫[c to d] ∫[h1(y) to h2(y)] f(x,y) dx dy |
How do you switch order of integration? | Sketch the region, rewrite the bounds, and reverse the order carefully. |
What is surface area for z=f(x,y)? | S = ∬R sqrt(1 + (fx)^2 + (fy)^2) dA |
What is a triple integral? | An integral of the form ∭E f(x,y,z) dV over a solid E. |
Volume of a solid using a triple integral | V = ∭E 1 dV |
General idea of iterated triple integrals | Write the triple integral as repeated one-variable integrals in an order that matches the region. |
Polar coordinate conversion: x | x = r cos(theta) |
Polar coordinate conversion: y | y = r sin(theta) |
Polar identity | x^2 + y^2 = r^2 |
Polar angle relation | tan(theta) = y/x |
Area element in polar coordinates | dA = r dr dtheta |
Double integral in polar coordinates | ∬R f(x,y) dA = ∬R f(r cos(theta), r sin(theta)) r dr dtheta |
Circle x^2 + y^2 = a^2 in polar | r = a |
Theta range for full circle | 0 ≤ theta ≤ 2pi |
Theta range for first quadrant | 0 ≤ theta ≤ pi/2 |
Cylindrical coordinate conversion: x | x = r cos(theta) |
Cylindrical coordinate conversion: y | y = r sin(theta) |
Cylindrical coordinate conversion: z | z = z |
Cylindrical identity | x^2 + y^2 = r^2 |
Volume element in cylindrical coordinates | dV = r dz dr dtheta |
When should cylindrical coordinates be used? | For solids with circular symmetry around the z-axis. |
Spherical coordinate conversion: x | x = rho sin(phi) cos(theta) |
Spherical coordinate conversion: y | y = rho sin(phi) sin(theta) |
Spherical coordinate conversion: z | z = rho cos(phi) |
Spherical identity | x^2 + y^2 + z^2 = rho^2 |
Volume element in spherical coordinates | dV = rho^2 sin(phi) drho dphi dtheta |
Meaning of rho in spherical coordinates | Distance from the origin. |
Meaning of theta in spherical coordinates | Angle in the xy-plane. |
Meaning of phi in spherical coordinates | Angle measured down from the positive z-axis. |
Standard theta range in spherical coordinates | 0 ≤ theta ≤ 2pi |
Standard phi range in spherical coordinates | 0 ≤ phi ≤ pi |
When should spherical coordinates be used? | For spheres, cones, and solids centered at the origin. |
Change of variables formula for double integrals | ∬R f(x,y) dA = ∬S f(x(u,v), y(u,v)) |∂(x,y)/∂(u,v)| du dv |
Jacobian in two variables | ∂(x,y)/∂(u,v) = | xu xv ; yu yv | |
Change of variables formula for triple integrals | ∭E f(x,y,z) dV = ∭G f(x(u,v,w), y(u,v,w), z(u,v,w)) |∂(x,y,z)/∂(u,v,w)| du dv dw |
Jacobian in three variables | ∂(x,y,z)/∂(u,v,w) = determinant of the 3x3 matrix of partial derivatives |
Jacobian for polar coordinates | |∂(x,y)/∂(r,theta)| = r |
Jacobian for cylindrical coordinates | |∂(x,y,z)/∂(r,theta,z)| = r |
Jacobian for spherical coordinates | |∂(x,y,z)/∂(rho,phi,theta)| = rho^2 sin(phi) |
How do you choose between polar, cylindrical, and spherical coordinates? | Use polar for circles/disks, cylindrical for circular solids around the z-axis, and spherical for spheres or origin-centered solids. |
What should you always include when changing coordinates? | The Jacobian. |
What should you do before changing order of integration? | Sketch the region. |
What should you do before setting up a multiple integral? | Understand the region and choose the best coordinate system/order. |
Most important optimization checklist | Find interior critical points, check boundaries or constraints, and compare values. |
Most important directional derivative checklist | Find the gradient and make sure the direction vector is a unit vector. |
Most important multiple integrals checklist | Sketch the region, choose the best order or coordinates, and include the Jacobian. |