12 Applications of Multivariable Calculus Free Online FlashCards

What is a level curve of $f(x,y)$?

A level curve is the set of points satisfying $f(x,y)=c$ for a constant $c$. It represents locations where the scalar function has the same value.

What does Green’s theorem convert?

Green’s theorem states $\oint_C P\,dx+Q\,dy=\iint_R(Q_x-P_y)\,dA$ for a positively oriented planar boundary.

What is the tangent-plane formula for $z=f(x,y)$?

The tangent plane is $z-z_0=f_x(x_0,y_0)(x-x_0)+f_y(x_0,y_0)(y-y_0)$.

How are velocity and acceleration obtained from $\mathbf r(t)$?

For $\mathbf r(t)=\langle x(t),y(t),z(t)\rangle$, velocity is $\mathbf r'(t)$ and acceleration is $\mathbf r''(t)$.

How is distance traveled along a space curve calculated?

The distance traveled from $t=a$ to $t=b$ is $L=\int_a^b\|\mathbf v(t)\|\,dt$, where $\mathbf v(t)=\mathbf r'(t)$.

What area element is used in polar coordinates?

In polar coordinates, $dA=r\,dr\,d\theta$. The factor $r$ is the Jacobian that accounts for coordinate stretching.

How are a lamina’s center-of-mass coordinates computed?

For a lamina, the center of mass is $\bar x=\frac{M_y}{m}$ and $\bar y=\frac{M_x}{m}$.

What does positive divergence indicate?

For $\mathbf F=\langle P,Q,R\rangle$, $\nabla\cdot\mathbf F=P_x+Q_y+R_z$. Positive divergence indicates local expansion or a source.

What characterizes a conservative vector field?

A conservative field has $\mathbf F=\nabla f$ for some potential function $f$. Its line integral depends only on the endpoints.

What is the parametrized formula for a scalar line integral?

The scalar line integral is $\int_C f\,ds=\int_a^b f(\mathbf r(t))\|\mathbf r'(t)\|\,dt$. It accumulates a scalar quantity along a curve.

What does flux through an oriented surface measure?

Flux through an oriented surface is $\iint_S\mathbf F\cdot\mathbf n\,dS$. It measures how much of the field crosses the surface.

What does the divergence theorem relate?

The divergence theorem states $\iint_{\partial E}\mathbf F\cdot\mathbf n\,dS=\iiint_E\nabla\cdot\mathbf F\,dV$.